P7248 [BalticOI 2012] Brackets (Day1)

A valid bracket sequence is defined as follows: - `()` and `[]` are valid bracket sequences. - If $A$ is a valid bracket sequence, then $(A)$ and $[A]$ are also valid bracket sequences. - If $A$ and $B$ are both valid bracket sequences, then $AB$ is also a valid bracket sequence. For a valid bracket sequence that contains at least one pair of square brackets, we can replace both `[` and `]` with `(`, and then we may obtain an invalid bracket sequence. For example, `((` and `((((()))` are both invalid bracket sequences. The former can be transformed from the valid bracket sequence `\[]`. The latter can be transformed from the following four valid bracket sequences: `\[]((()))`, `(\[](()))`, `((\[]()))`, and `(((\[])))`. Now you are given an invalid bracket sequence. Find how many valid bracket sequences can become the given sequence after replacing the square brackets in them with `(`.

The first line contains a positive integer $n$, which denotes the length of the given invalid bracket sequence. The second line contains a string of length $n$, representing the given sequence, which contains only `(` and `)`.

Output the number of valid bracket sequences that meet the requirement modulo $10^9+9$.

**Sample Explanation #1.** There are two valid bracket sequences that satisfy the condition: `\[]()` and `([])`. **Constraints.** - For 20% of the testdata, $n \leq 50$. - For 50% of the testdata, $n \leq 1000$. - For 100% of the testdata, $2 \leq n \leq 30000$. **Notes.** Translated from [BalticOI 2012 Day1 T1. Brackets](http://www.boi2012.lv/data/day1/eng/brackets.pdf). Translated by ChatGPT 5