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F - Integer Division

Score : $500$ points

Problem Statement

You are given a positive integer $X$ with $N$ digits in decimal representation. None of the digits of $X$ is $0$.
For a subset $S$ of $\lbrace 1,2, \ldots, N-1 \rbrace $, let $f(S)$ be defined as follows.

See the decimal representation of $X$ as a string of length $N$, and decompose it into $|S| + 1$ strings by splitting it between the $i$-th and $(i + 1)$-th characters if and only if $i \in S$.
Then, see these $|S| + 1$ strings as integers in decimal representation, and let $f(S)$ be the product of these $|S| + 1$ integers.

There are $2^{N-1}$ subsets of $\lbrace 1,2, \ldots, N-1 \rbrace $, including the empty set, which can be $S$. Find the sum of $f(S)$ over all these $S$, modulo $998244353$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $X$ has $N$ digits in decimal representation, none of which is $0$.
• All values in the input are integers.

Input

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

$N$

$X$

Output

Print the answer.

3
234

418

For $S = \emptyset$, we have $f(S) = 234$.
For $S = \lbrace 1 \rbrace$, we have $f(S) = 2 \times 34 = 68$.
For $S = \lbrace 2 \rbrace$, we have $f(S) = 23 \times 4 = 92$.
For $S = \lbrace 1, 2 \rbrace$, we have $f(S) = 2 \times 3 \times 4 = 24$.
Thus, you should print $234 + 68 + 92 + 24 = 418$.

4
5915

17800

9
998244353

258280134