Build From Suffixes
Description
You are given an integer $n$ and a sequence $a$ of $n-1$ integers, each element is either $0$ or $1$.
You are asked to build a string of length $n$ such that:
- each letter is one of "abcd";
- if $a_i=1$, then the $i$-th suffix of the string is lexicographically smaller than the $(i+1)$-th suffix;
- if $a_i=0$, then the $i$-th suffix of the string is lexicographically greater than the $(i+1)$-th suffix.
You will be asked $q$ queries of form:
- $i$ ($1 \le i \le n - 1$) — flip the value of $a_i$ (if $a_i=0$, then set $a_i$ to $1$, and vice versa).
After each query print the number of different strings that satisfy the given constraints modulo $998\,244\,353$.
The first line contains two integers $n$ and $q$ ($2 \le n \le 10^5$; $1 \le q \le 10^5$) — the length of the string and the number of queries.
The second line contains $n-1$ integers $a_1, a_2, \dots, a_{n-1}$ ($a_i \in \{0, 1\}$) — the constraints on suffixes of the string.
Each of the next $q$ lines contains a query: an integer $i$ ($1 \le i \le n - 1$) — flip the value of $a_i$ (if $a_i=0$, then set $a_i$ to $1$, and vice versa).
After each query print the number of different strings that satisfy the given constraints modulo $998\,244\,353$.
Input
The first line contains two integers $n$ and $q$ ($2 \le n \le 10^5$; $1 \le q \le 10^5$) — the length of the string and the number of queries.
The second line contains $n-1$ integers $a_1, a_2, \dots, a_{n-1}$ ($a_i \in \{0, 1\}$) — the constraints on suffixes of the string.
Each of the next $q$ lines contains a query: an integer $i$ ($1 \le i \le n - 1$) — flip the value of $a_i$ (if $a_i=0$, then set $a_i$ to $1$, and vice versa).
Output
After each query print the number of different strings that satisfy the given constraints modulo $998\,244\,353$.
Samples
2 2
0
1
1
6
10
3 4
0 0
1
2
1
2
20
10
14
20
10 10
0 0 0 1 1 1 0 1 0
4
1
8
4
2
9
5
6
6
8
1815
3201
2710
2776
2290
1644
2668
1271
2668
2436
Note
The $i$-th suffix of a string is a continuous substring that starts from the $i$-th position and ends in the last position.
A string $a$ is lexicographically smaller than a string $b$ if and only if one of the following holds:
- $a$ is a prefix of $b$, but $a \ne b$;
- in the first position where $a$ and $b$ differ, the string $a$ has a letter that appears earlier in the alphabet than the corresponding letter in $b$.
Two strings $a$ and $b$ of length $n$ differ if there exists a position $i$ such that $a_i \neq b_i$.