2019/finals/strings_as_a_service.md

Carlos has been working in technology so long that he's starting to feel a bit
burnt out. Hoping to rejuvenate himself, Carlos has been seeking out more
artistic opportunities.

Yamaha, the well-known creator of musical apparatus, has approached Carlos
with a request that might be right up his alley: they'd like him to design a
brand new instrument. Immediately, Carlos knows what to do.

_ "You may have seen Pat Metheny's 42-string guitar, but that's nothing
compared to what we're going to make together." _

Carlos presents his plan for a 1,000-string guitar, complete with programmatic
tuning so that you don't need to turn 1,000 knobs by hand. Yamaha's market
research suggests that these sorts of guitars would be great for playing
palindromic chords, chords where the first string plays the same note as the
last string, the second string plays the same note as the second-to-last
string, and so on. Carlos is quickly tasked with developing default tunings
for the strings so that the guitars are ready to play right out of the box.

For various integers **K**, Carlos wants to find a set of at most 1,000
strings on which exactly **K** distinct palindromic chords can be played. The
guitar's strings are arranged in a line, and each one must be tuned to a note
from the set {A, B, C, D, E, F, G}. A chord is then played by strumming a
contiguous subset of 1 or more strings. Two chords are considered to be
distinct if there is at least one string that is used in one chord but not the
other; chords involving the same notes but different strings are considered
different.

For example, if **K** = 9, a set of 7 strings could be tuned to the notes C,
A, B, B, A, G, E in order from left to right. You can play 7 different
palindromic chords by strumming single strings, the chord BB by strumming the
3rd and 4th strings, and the chord ABBA by strumming the 2nd, 3rd, 4th, and
5th strings. This is a total of 9 distinct palindromic chords. **

Output any non-empty string of valid musical notes, with length at most 1,000,
representing the tunings of sequential strings. An aspiring musician must be
able to play exactly **K** distinct palindromic chords on these strings. It's
guaranteed that there is at least one valid output for each possible valid
input.

### Input

Input begins with an integer **T**, the number of tunings that Carlos needs to
figure out.  
For each tuning, there is a single line containing the integer **K**.

### Output

For the _i_th tuning, print a line containing "Case #_i_: " followed by a
string of up to 1,000 characters representing a tuning of strings as described
above on which exactly **K** distinct palindromic chords can be played.

### Constraints

1 ≤ **T** ≤ 500  
1 ≤ **K** ≤ 100,000  

### Explanation of Sample

In the first case, "ACE" is a valid output as it contains exactly 3
palindromes: "A", "C", and "E". On the other hand, "DAD" would not be valid as
it contains 4 palindromes.

In the second case, "GAGA" is a valid output as it contains exactly 6
palindromes: "G", "A", "G", "A", "GAG", and "AGA".

**_Note that other outputs would also be accepted for each sample case._**