A regular bracket-sequence is a string of characters consisting only of opening and closing brackets, and satisfying the following conditions:
An empty string is a regular bracket-sequence.
If $A$ is a regular bracket-sequence, then ($A$), [$A$] and {$A$} are also regular bracket-sequences.
If $A$ and $B$ are regular bracket-sequences, then $AB$ is also a regular bracket-sequence.
For example, the sequences “[({})]”, “[](){}” and “[{}]()[{}]” are regular, but the sequences “[({{([”, “[]({)}” and “[{}])([{}]” are not.
Ivica has found a string which looks like it could be a regular bracket-sequence. Some of the characters have become smudged and illegible, and could have been any character.
Write a program that calculates how many ways the illegible characters in the string can be replaced by brackets so that the result is a regular bracket-sequence. This number can be very large, so output only its last $5$ digits.
The first line contains an even integer $N$ ($2 \le N \le 200$), the length of the string. The second line contains the string. Illegible characters are represented by the ‘?’ character.
Output the number of regular bracket-sequences the string could have read.
Sample Input 1 | Sample Output 1 |
---|---|
6 ()()() |
1 |
Sample Input 2 | Sample Output 2 |
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10 (?([?)]?}? |
3 |
Sample Input 3 | Sample Output 3 |
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16 ???[???????]???? |
92202 |