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Problem J
Balanced Tree Path

You are given a tree where each node is annotated with a character from ()[]{}. A path is a sequence of one or more nodes where no node is repeated and every pair of adjacent nodes is connected with an edge. A path is balanced if the characters at each node, when concatenated, form a balanced string. A string is balanced if it satisfies the following definition:

  • An empty string is balanced.

  • If $s$ is a balanced string, then ($s$), [$s$], and {$s$} are balanced strings.

  • if $a$ and $b$ are balanced strings, then $ab$ ($a$ concatenated with $b$) is a balanced string.

Compute the number of balanced paths over the entire tree.

Input

The first line of input contains a single integer $n$ ($2 \le n \le 5 \cdot 10^3$).

The next line contains a string of $n$ characters, where each character is one of ()[]{}.

Each of the next $n-1$ lines contains two integers, $u$ and $v$ ($1 \le u < v \le n$), indicating that nodes $u$ and $v$ are connected with an edge. It is guaranteed the graph is a tree.

Output

Output a single integer, which is the number of balanced paths over the entire tree.

Sample Input 1 Sample Output 1
4
()()
1 2
2 3
3 4
4
Sample Input 2 Sample Output 2
4
[[]]
1 2
2 3
3 4
2
Sample Input 3 Sample Output 3
6
([]{})
1 2
2 3
3 4
4 5
5 6
4

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