2015/finals/fox_hawks.md

Mr. Fox always puts aside some time on the weekends to practice his falconry.
Mr. Fox owns **N** hawks, numbered from 1 to **N**. While numbering is
somewhat impersonal, it quickly becomes infeasible to name each hawk
individually when you have as many hawks as Mr. Fox.

Every year, the local falconer club hosts a festival for falconers from across
the nation. Mr. Fox shows off some of his hawks at each festival, and this
year is no different. Selecting a set of hawks to display is not a
straightforward task however. Hawks can be temperamental creatures, and
they'll refuse to perform if they don't like the situation they find
themselves in. Luckily, after careful study, Mr. Fox has been able to capture
the hawks' preferences in a simple boolean expression.

For example, let's say Mr. Fox has 4 hawks. Hawk 1 will only perform if some
other hawk is present. Hawks 2 and 3 will only perform if hawks 1 or 4 are
present. Hawk 4 is much more easy-going and will perform in all situations. We
can express these preferences with the following expression:

    ((1 & (2 | 3)) | 4)

Each number is a boolean variable indicating whether or not Mr. Fox brings
that hawk. If the expression is satisfied, then all of the hawks he brings
will perform. If the expression is not satisfied, the hawks will be moody and
that means no blue ribbons for Mr. Fox.

Mr. Fox is keen not to bore his audience, so he always brings a different set
of hawks each year. This is the **K**th annual festival, so he would like to
bring the set of performing hawks with the **K**th lowest value. Mr. Fox
defines the value of a set of hawks as follows: the empty set has a value of
0, and hawk **i** adds 2**i** to the value of a set. So with 3 hawks, the sets
in increasing order are:

    {1}
    {2}
    {1, 2}
    {3}
    {1, 3}
    {2, 3}
    {1, 2, 3}

Note that Mr. Fox always brings a non-empty set of hawks.

### Input

Input begins with an integer **T**, the number of festivals under
consideration. For each festival, there is first a line containing the space-
separated integers **N** and **K**. The next line contains the boolean
expression encoding the hawks' preferences.

### Output

For the **i**th festival, print a line containing "Case #**i**: " followed by
value of the set of hawks that Mr. Fox brings modulo 109+7.

### Constraints

1 ≤ **T** ≤ 20  
1 ≤ **N** ≤ 200,000  
1 ≤ **K** ≤ 1018  
Expressions contain no more than 2,500,000 characters each.  
It is guaranteed that there are at least **K** sets of performing hawks.  

The boolean expression adheres to the following grammar:

    [expression] ::= "(" "~" [expression] ")" | "(" [expression] [binary-operator] [expression] ")" | [variable]
    [binary-operator] ::= "|" | "^" | "&"
    [variable] ::= [digit] | [digit] [variable]
    [digit] ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9"

Each hawk appears in the boolean expression exactly once.  
Whitespace may appear arbitrarily in the expression (except within variables)
to improve readability.  

### Explanation of Sample

In the first and second cases, the first 4 performing sets, in order, are {1,
2}, {1, 3}, {1, 2, 3}, and {4}, with values of 6, 10, 14, and 16 respectively.