Description:
Chiyuu has a bracket sequence$$$^\dagger$$$ $$$s$$$ of length $$$n$$$. Let $$$k$$$ be the minimum number of characters that Chiyuu has to remove from $$$s$$$ to make $$$s$$$ balanced$$$^\ddagger$$$.
Now, Koxia wants you to count the number of ways to remove $$$k$$$ characters from $$$s$$$ so that $$$s$$$ becomes balanced, modulo $$$998\,244\,353$$$.
Note that two ways of removing characters are considered distinct if and only if the set of indices removed is different.
$$$^\dagger$$$ A bracket sequence is a string containing only the characters "(" and ")".
$$$^\ddagger$$$ A bracket sequence is called balanced if one can turn it into a valid math expression by adding characters + and 1. For example, sequences (())(), (), (()(())) and the empty string are balanced, while )(, ((), and (()))( are not.
Input Format:
The first line of input contains a string $$$s$$$ ($$$1 \leq |s| \leq 5 \cdot {10}^5$$$) — the bracket sequence.
It is guaranteed that $$$s$$$ only contains the characters "(" and ")".
Output Format:
Output a single integer — the number of ways to remove $$$k$$$ characters from $$$s$$$ so that $$$s$$$ becomes balanced, modulo $$$998\,244\,353$$$.
Note:
In the first test case, it can be proved that the minimum number of characters that Chiyuu has to remove is $$$2$$$. There are $$$4$$$ ways to remove $$$2$$$ characters to make $$$s$$$ balanced as follows. Deleted characters are noted as red.
- $$$\texttt{(} \color{Red}{\texttt{)}} \texttt{)} \color{Red}{\texttt{(}} \texttt{(} \texttt{)}$$$,
- $$$\texttt{(} \texttt{)} \color{Red}{\texttt{)}} \color{Red}{\texttt{(}} \texttt{(} \texttt{)}$$$,
- $$$\texttt{(} \color{Red}{\texttt{)}} \texttt{)} \texttt{(} \color{Red}{\texttt{(}} \texttt{)}$$$,
- $$$\texttt{(} \texttt{)} \color{Red}{\texttt{)}} \texttt{(} \color{Red}{\texttt{(}} \texttt{)}$$$.
In the second test case, the only way to make $$$s$$$ balanced is by deleting the only character to get an empty bracket sequence, which is considered balanced.