A11317. Counting Is Fun

You are given a binary $^\dagger$ pattern $p$ of length $n$ .

A binary string $q$ of the same length $n$ is called good if for every $i$ ( $1 \leq i \leq n$ ), there exist indices $l$ and $r$ such that:

- $1 \leq l \leq i \leq r \leq n$ , and
- $p_i$ is a mode $^\ddagger$ of the string $q_lq_{l+1}\ldots q_r$ .

Count the number of good binary strings modulo $998\,244\,353$ .

$^\dagger$ A binary string is a string that only consists of characters $\mathtt{0}$ and $\mathtt{1}$ .

$^\ddagger$ Character $c$ is a mode of string $t$ of length $m$ if the number of occurrences of $c$ in $t$ is at least $\lceil \frac{m}{2} \rceil$ . For example, $\mathtt{0}$ is a mode of $\mathtt{010}$ , $\mathtt{1}$ is not a mode of $\mathtt{010}$ , and both $\mathtt{0}$ and $\mathtt{1}$ are modes of $\mathtt{011010}$ .

The first line of input contains a single integer $n$ ( $1 \le n \le 10^5$ ) — the length of the binary string $p$ .

The second line of input contains a binary string $p$ of length $n$ consisting of characters 0 and 1.

Output the number of good strings modulo $998\,244\,353$ .

In the second example, the good strings are

- $\mathtt{010}$ ;
- $\mathtt{011}$ ;
- $\mathtt{101}$ ;
- $\mathtt{110}$ ;
- $\mathtt{111}$ .