Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.
from typing import List

def factorize(n: int) -> List[int]:
    """ Return list of prime factors of given integer in the order from smallest to largest.
    Each of the factors should be listed number of times corresponding to how many times it appeares in factorization.
    Input number should be equal to the product of all factors
    >>> factorize(8)
    [2, 2, 2]
    >>> factorize(25)
    [5, 5]
    >>> factorize(70)
    [2, 5, 7]
    """

Response:

Output:
    import math
    fact = []
    i = 2
    while i <= int(math.sqrt(n) + 1):
        if n % i == 0:
            fact.append(i)
            n //= i
        else:
            i += 1

    if n > 1:
        fact.append(n)
    return fact


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.
from typing import List

def remove_duplicates(numbers: List[int]) -> List[int]:
    """ From a list of integers, remove all elements that occur more than once.
    Keep order of elements left the same as in the input.
    >>> remove_duplicates([1, 2, 3, 2, 4])
    [1, 3, 4]
    """

Response:

Output:
    import collections
    c = collections.Counter(numbers)
    return [n for n in numbers if c[n] <= 1]


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def flip_case(string: str) -> str:
    """ For a given string, flip lowercase characters to uppercase and uppercase to lowercase.
    >>> flip_case('Hello')
    'hELLO'
    """

Response:

Output: return string.swapcase()


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.
from typing import List

def concatenate(strings: List[str]) -> str:
    """ Concatenate list of strings into a single string
    >>> concatenate([])
    ''
    >>> concatenate(['a', 'b', 'c'])
    'abc'
    """

Response:

Output: return ''.join(strings)


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.
from typing import List

def filter_by_prefix(strings: List[str], prefix: str) -> List[str]:
    """ Filter an input list of strings only for ones that start with a given prefix.
    >>> filter_by_prefix([], 'a')
    []
    >>> filter_by_prefix(['abc', 'bcd', 'cde', 'array'], 'a')
    ['abc', 'array']
    """

Response:

Output: return [x for x in strings if x.startswith(prefix)]


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def get_positive(l: list):
    """Return only positive numbers in the list.
    >>> get_positive([-1, 2, -4, 5, 6])
    [2, 5, 6]
    >>> get_positive([5, 3, -5, 2, -3, 3, 9, 0, 123, 1, -10])
    [5, 3, 2, 3, 9, 123, 1]
    """

Response:

Output: return [e for e in l if e > 0]


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def is_prime(n):
    """Return true if a given number is prime, and false otherwise.
    >>> is_prime(6)
    False
    >>> is_prime(101)
    True
    >>> is_prime(11)
    True
    >>> is_prime(13441)
    True
    >>> is_prime(61)
    True
    >>> is_prime(4)
    False
    >>> is_prime(1)
    False
    """

Response:

Output:
    if n < 2:
        return False
    for k in range(2, n - 1):
        if n % k == 0:
            return False
    return True


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.
import math

def poly(xs: list, x: float):
    """
    Evaluates polynomial with coefficients xs at point x.
    return xs[0] + xs[1] * x + xs[1] * x^2 + .... xs[n] * x^n
    """
    return sum([coeff * math.pow(x, i) for i, coeff in enumerate(xs)])

def find_zero(xs: list):
    """ xs are coefficients of a polynomial.
    find_zero find x such that poly(x) = 0.
    find_zero returns only only zero point, even if there are many.
    Moreover, find_zero only takes list xs having even number of coefficients
    and largest non zero coefficient as it guarantees
    a solution.
    >>> round(find_zero([1, 2]), 2) # f(x) = 1 + 2x
    -0.5
    >>> round(find_zero([-6, 11, -6, 1]), 2) # (x - 1) * (x - 2) * (x - 3) = -6 + 11x - 6x^2 + x^3
    1.0
    """

Response:

Output:
    begin, end = -1., 1.
    while poly(xs, begin) * poly(xs, end) > 0:
        begin *= 2.0
        end *= 2.0
    while end - begin > 1e-10:
        center = (begin + end) / 2.0
        if poly(xs, center) * poly(xs, begin) > 0:
            begin = center
        else:
            end = center
    return begin


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def sort_third(l: list):
    """This function takes a list l and returns a list l' such that
    l' is identical to l in the indicies that are not divisible by three, while its values at the indicies that are divisible by three are equal
    to the values of the corresponding indicies of l, but sorted.
    >>> sort_third([1, 2, 3])
    [1, 2, 3]
    >>> sort_third([5, 6, 3, 4, 8, 9, 2])
    [2, 6, 3, 4, 8, 9, 5]
    """

Response:

Output:
    l = list(l)
    l[::3] = sorted(l[::3])
    return l


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def unique(l: list):
    """Return sorted unique elements in a list
    >>> unique([5, 3, 5, 2, 3, 3, 9, 0, 123])
    [0, 2, 3, 5, 9, 123]
    """

Response:

Output: return sorted(list(set(l)))


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def max_element(l: list):
    """Return maximum element in the list.
    >>> max_element([1, 2, 3])
    3
    >>> max_element([5, 3, -5, 2, -3, 3, 9, 0, 123, 1, -10])
    123
    """

Response:

Output:
    m = l[0]
    for e in l:
        if e > m:
            m = e
    return m


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def fizz_buzz(n: int):
    """Return the number of times the digit 7 appears in integers less than n which are divisible by 11 or 13.
    >>> fizz_buzz(50)
    0
    >>> fizz_buzz(78)
    2
    >>> fizz_buzz(79)
    3
    """

Response:

Output:
    ns = []
    for i in range(n):
        if i % 11 == 0 or i % 13 == 0:
            ns.append(i)
    s = ''.join(list(map(str, ns)))
    ans = 0
    for c in s:
        ans += (c == '7')
    return ans


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def sort_even(l: list):
    """This function takes a list l and returns a list l' such that
    l' is identical to l in the odd indicies, while its values at the even indicies are equal
    to the values of the even indicies of l, but sorted.
    >>> sort_even([1, 2, 3])
    [1, 2, 3]
    >>> sort_even([5, 6, 3, 4])
    [3, 6, 5, 4]
    """

Response:

Output:
    evens = l[::2]
    odds = l[1::2]
    evens.sort()
    ans = []
    for e, o in zip(evens, odds):
        ans.extend([e, o])
    if len(evens) > len(odds):
        ans.append(evens[-1])
    return ans


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def encode_cyclic(s: str):
    """
    returns encoded string by cycling groups of three characters.
    """
    # split string to groups. Each of length 3.
    groups = [s[(3 * i):min((3 * i + 3), len(s))] for i in range((len(s) + 2) // 3)]
    # cycle elements in each group. Unless group has fewer elements than 3.
    groups = [(group[1:] + group[0]) if len(group) == 3 else group for group in groups]
    return "".join(groups)

def decode_cyclic(s: str):
    """
    takes as input string encoded with encode_cyclic function. Returns decoded string.
    """

Response:

Output: return encode_cyclic(encode_cyclic(s))


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def prime_fib(n: int):
    """
    prime_fib returns n-th number that is a Fibonacci number and it's also prime.
    >>> prime_fib(1)
    2
    >>> prime_fib(2)
    3
    >>> prime_fib(3)
    5
    >>> prime_fib(4)
    13
    >>> prime_fib(5)
    89
    """

Response:

Output:
    import math

    def is_prime(p):
        if p < 2:
            return False
        for k in range(2, min(int(math.sqrt(p)) + 1, p - 1)):
            if p % k == 0:
                return False
        return True
    f = [0, 1]
    while True:
        f.append(f[-1] + f[-2])
        if is_prime(f[-1]):
            n -= 1
        if n == 0:
            return f[-1]


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def triples_sum_to_zero(l: list):
    """
    triples_sum_to_zero takes a list of integers as an input.
    it returns True if there are three distinct elements in the list that
    sum to zero, and False otherwise.

    >>> triples_sum_to_zero([1, 3, 5, 0])
    False
    >>> triples_sum_to_zero([1, 3, -2, 1])
    True
    >>> triples_sum_to_zero([1, 2, 3, 7])
    False
    >>> triples_sum_to_zero([2, 4, -5, 3, 9, 7])
    True
    >>> triples_sum_to_zero([1])
    False
    """

Response:

Output:
    for i in range(len(l)):
        for j in range(i + 1, len(l)):
            for k in range(j + 1, len(l)):
                if l[i] + l[j] + l[k] == 0:
                    return True
    return False


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def car_race_collision(n: int):
    """
    Imagine a road that's a perfectly straight infinitely long line.
    n cars are driving left to right;  simultaneously, a different set of n cars
    are driving right to left.   The two sets of cars start out being very far from
    each other.  All cars move in the same speed.  Two cars are said to collide
    when a car that's moving left to right hits a car that's moving right to left.
    However, the cars are infinitely sturdy and strong; as a result, they continue moving
    in their trajectory as if they did not collide.

    This function outputs the number of such collisions.
    """

Response:

Output: return n**2


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def incr_list(l: list):
    """Return list with elements incremented by 1.
    >>> incr_list([1, 2, 3])
    [2, 3, 4]
    >>> incr_list([5, 3, 5, 2, 3, 3, 9, 0, 123])
    [6, 4, 6, 3, 4, 4, 10, 1, 124]
    """

Response:

Output: return [(e + 1) for e in l]


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def pairs_sum_to_zero(l):
    """
    pairs_sum_to_zero takes a list of integers as an input.
    it returns True if there are two distinct elements in the list that
    sum to zero, and False otherwise.
    >>> pairs_sum_to_zero([1, 3, 5, 0])
    False
    >>> pairs_sum_to_zero([1, 3, -2, 1])
    False
    >>> pairs_sum_to_zero([1, 2, 3, 7])
    False
    >>> pairs_sum_to_zero([2, 4, -5, 3, 5, 7])
    True
    >>> pairs_sum_to_zero([1])
    False
    """

Response:

Output:
    for i, l1 in enumerate(l):
        for j in range(i + 1, len(l)):
            if l1 + l[j] == 0:
                return True
    return False


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def change_base(x: int, base: int):
    """Change numerical base of input number x to base.
    return string representation after the conversion.
    base numbers are less than 10.
    >>> change_base(8, 3)
    '22'
    >>> change_base(8, 2)
    '1000'
    >>> change_base(7, 2)
    '111'
    """

Response:

Output:
    ret = ""
    while x > 0:
        ret = str(x % base) + ret
        x //= base
    return ret


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def triangle_area(a, h):
    """Given length of a side and high return area for a triangle.
    >>> triangle_area(5, 3)
    7.5
    """

Response:

Output: return a * h / 2.0


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def fib4(n: int):
    """The Fib4 number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
    fib4(0) -> 0
    fib4(1) -> 0
    fib4(2) -> 2
    fib4(3) -> 0
    fib4(n) -> fib4(n-1) + fib4(n-2) + fib4(n-3) + fib4(n-4).
    Please write a function to efficiently compute the n-th element of the fib4 number sequence.  Do not use recursion.
    >>> fib4(5)
    4
    >>> fib4(6)
    8
    >>> fib4(7)
    14
    """

Response:

Output:
    results = [0, 0, 2, 0]
    if n < 4:
        return results[n]

    for _ in range(4, n + 1):
        results.append(results[-1] + results[-2] + results[-3] + results[-4])
        results.pop(0)

    return results[-1]


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Complete the implementation of the function below.
from typing import List

def has_close_elements(numbers: List[float], threshold: float) -> bool:
    """ Check if in given list of numbers, are any two numbers closer to each other than
    given threshold.
    >>> has_close_elements([1.0, 2.0, 3.0], 0.5)
    False
    >>> has_close_elements([1.0, 2.8, 3.0, 4.0, 5.0, 2.0], 0.3)
    True
    """

Response:
Step 1: Use a nested loop to compare each number in the list with every other number.
Step 2: Calculate the absolute difference between each pair of distinct elements.
Step 3: Return True if any difference is less than the threshold; otherwise, return False after checking all pairs.
Final Answer: \boxed{
    for idx, elem in enumerate(numbers):
        for idx2, elem2 in enumerate(numbers):
            if idx != idx2:
                distance = abs(elem - elem2)
                if distance < threshold:
                    return True

    return False
} ####

Question: Complete the implementation of the function below.
from typing import List

def separate_paren_groups(paren_string: str) -> List[str]:
    """ Input to this function is a string containing multiple groups of nested parentheses. Your goal is to
    separate those group into separate strings and return the list of those.
    Separate groups are balanced (each open brace is properly closed) and not nested within each other
    Ignore any spaces in the input string.
    >>> separate_paren_groups('( ) (( )) (( )( ))')
    ['()', '(())', '(()())']
    """

Response:
Step 1: Initialize a `result` list for groups, a `current_string` buffer, and a `current_depth` counter to track nesting levels.
Step 2: Iterate through the string, incrementing depth for `'('` and decrementing for `')'`, while appending brackets to the buffer (ignoring spaces).
Step 3: When `current_depth` returns to zero, a balanced group is complete; join the buffer into a string, add it to `result`, and reset the buffer.
Final Answer: \boxed{
    result = []
    current_string = []
    current_depth = 0

    for c in paren_string:
        if c == '(':
            current_depth += 1
            current_string.append(c)
        elif c == ')':
            current_depth -= 1
            current_string.append(c)

            if current_depth == 0:
                result.append(''.join(current_string))
                current_string.clear()

    return result
} ####

Question: Complete the implementation of the function below.

def truncate_number(number: float) -> float:
    """ Given a positive floating point number, it can be decomposed into
    and integer part (largest integer smaller than given number) and decimals
    (leftover part always smaller than 1).

    Return the decimal part of the number.
    >>> truncate_number(3.5)
    0.5
    """

Response:
Step 1: Use the modulo operator (`%`) with `1.0` to isolate the remainder after division by 1, which represents the decimal component.
Step 2: Alternatively, subtract the integer cast of the number (`int(number)`) from the original float to extract the leftover fractional part.
Final Answer: \boxed{
    return number % 1.0
} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Complete the implementation of the function below.

def median(l: list):
    """Return median of elements in the list l.
    >>> median([3, 1, 2, 4, 5])
    3
    >>> median([-10, 4, 6, 1000, 10, 20])
    15.0
    """

Response:

Output:
    l = sorted(l)
    if len(l) % 2 == 1:
        return l[len(l) // 2]
    else:
        return (l[len(l) // 2 - 1] + l[len(l) // 2]) / 2.0