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G - Banned Substrings

Score : $550$ points

Problem Statement

You are given a set $S=\lbrace s _ 1,s _ 2,\ldots,s _ M\rbrace$ of non-empty strings of length at most $6$ consisting of a and b. Find the number of strings $T$ of length $N$ consisting of a and b that satisfy the following condition:

• $T$ does not contain $s _ i$ as a consecutive substring for any $s _ i\in S$.

Since the answer can be enormous, find it modulo $998244353$.

Constraints

• $1\leq N\leq10^{18}$
• $1\leq M\leq126$
• $N$ and $M$ are integers.
• $s _ i$ is a non-empty string of length at most $6$ consisting of a and b.
• $s _ i\neq s _ j\ (1\leq i\lt j\leq M)$

Input

The input is given from Standard Input in the following format:

$N$ $M$

$s _ 1$

$s _ 2$

$\vdots$

$s _ M$

Output

Print the answer modulo $998244353$ in a single line.

4 3
aab
bbab
abab

10

There are $10$ strings of length $4$ consisting of a and b that do not contain aab, bbab, or abab as consecutive substrings: aaaa, abaa, abba, abbb, baaa, baba, babb, bbaa, bbba, and bbbb. Thus, you should print $10$.

20 1
aa

17711

1000000007 28
bbabba
bbbbaa
aabbab
bbbaba
baaabb
babaab
bbaaba
aabaaa
aaaaaa
aabbaa
bbaaaa
bbaabb
bbabab
aababa
baaaba
ababab
abbaba
aabaab
ababaa
abbbba
baabaa
aabbbb
abbbab
baaaab
baabbb
ababbb
baabba
abaaaa

566756841

Print the answer modulo $998244353$.