Side Transmutations

Consider some set of distinct characters $$$A$$$ and some string $$$S$$$, consisting of exactly $$$n$$$ characters, where each character is present in $$$A$$$.

You are given an array of $$$m$$$ integers $$$b$$$ ($$$b_1 < b_2 < \dots < b_m$$$).

You are allowed to perform the following move on the string $$$S$$$:

1. Choose some valid $$$i$$$ and set $$$k = b_i$$$;
2. Take the first $$$k$$$ characters of $$$S = Pr_k$$$;
3. Take the last $$$k$$$ characters of $$$S = Su_k$$$;
4. Substitute the first $$$k$$$ characters of $$$S$$$ with the reversed $$$Su_k$$$;
5. Substitute the last $$$k$$$ characters of $$$S$$$ with the reversed $$$Pr_k$$$.

For example, let's take a look at $$$S =$$$ "abcdefghi" and $$$k = 2$$$. $$$Pr_2 =$$$ "ab", $$$Su_2 =$$$ "hi". Reversed $$$Pr_2 =$$$ "ba", $$$Su_2 =$$$ "ih". Thus, the resulting $$$S$$$ is "ihcdefgba".

The move can be performed arbitrary number of times (possibly zero). Any $$$i$$$ can be selected multiple times over these moves.

Let's call some strings $$$S$$$ and $$$T$$$ equal if and only if there exists such a sequence of moves to transmute string $$$S$$$ to string $$$T$$$. For the above example strings "abcdefghi" and "ihcdefgba" are equal. Also note that this implies $$$S = S$$$.

The task is simple. Count the number of distinct strings.

The answer can be huge enough, so calculate it modulo $$$998244353$$$.