F - Problem where +s Separate Digits

Score : $500$ points

Problem Statement

Given is a string $S$ consisting of digits from $1$ through $9$.
From this string $S$, let us make a formula $T$ by the following operations.

• Initially, let $T=S$.
• Choose a (possibly empty) set $A$ of different integers where each element is between $1$ and $|S|-1$ (inclusive).
• For each element $x$ in descending order, do the following.
  • Insert a + between the $x$-th and $(x+1)$-th characters of $T$.

For example, when $S=$ 1234 and $A= \lbrace 2,3 \rbrace$, we will have $T$= 12+3+4.

Consider evaluating all possible formulae $T$ obtained by these operations. Find the sum, modulo $998244353$, of the evaluations.

Constraints

• $1 \le |S| \le 2 \times 10^5$
• $S$ consists of 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Input

Input is given from Standard Input in the following format:

$S$

Output

Print the answer.

1234

1736

There are eight formulae that can be obtained as $T$: 1234, 123+4, 12+34, 12+3+4, 1+234, 1+23+4, 1+2+34, and 1+2+3+4.
The sum of the evaluations of these formulae is $1736$.

1

1

$S$ may have a length of $1$, in which case the only possible choice for $A$ is the empty set.

31415926535897932384626433832795

85607943

Be sure to find the sum modulo $998244353$.