Two Sorts

4000ms

524288K

Description:

Integers from $$$1$$$ to $$$n$$$ (inclusive) were sorted lexicographically (considering integers as strings). As a result, array $$$a_1, a_2, \dots, a_n$$$ was obtained.

Calculate value of $$$(\sum_{i = 1}^n ((i - a_i) \mod 998244353)) \mod 10^9 + 7$$$.

$$$x \mod y$$$ here means the remainder after division $$$x$$$ by $$$y$$$. This remainder is always non-negative and doesn't exceed $$$y - 1$$$. For example, $$$5 \mod 3 = 2$$$, $$$(-1) \mod 6 = 5$$$.

Input:

The first line contains the single integer $$$n$$$ ($$$1 \leq n \leq 10^{12}$$$).

Output:

Print one integer — the required sum.

Sample Input:

Sample Output:

Sample Input:

Sample Output:

994733045

Sample Input:

Sample Output:

978932159

Sample Input:

1000000000000

Sample Output:

289817887

Note:

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 $$$b$$$.

For example, $$$42$$$ is lexicographically smaller than $$$6$$$, because they differ in the first digit, and $$$4 < 6$$$; $$$42 < 420$$$, because $$$42$$$ is a prefix of $$$420$$$.

Let's denote $$$998244353$$$ as $$$M$$$.

In the first example, array $$$a$$$ is equal to $$$[1, 2, 3]$$$.

$$$(1 - 1) \mod M = 0 \mod M = 0$$$
$$$(2 - 2) \mod M = 0 \mod M = 0$$$
$$$(3 - 3) \mod M = 0 \mod M = 0$$$

As a result, $$$(0 + 0 + 0) \mod 10^9 + 7 = 0$$$

In the second example, array $$$a$$$ is equal to $$$[1, 10, 11, 12, 2, 3, 4, 5, 6, 7, 8, 9]$$$.

$$$(1 - 1) \mod M = 0 \mod M = 0$$$
$$$(2 - 10) \mod M = (-8) \mod M = 998244345$$$
$$$(3 - 11) \mod M = (-8) \mod M = 998244345$$$
$$$(4 - 12) \mod M = (-8) \mod M = 998244345$$$
$$$(5 - 2) \mod M = 3 \mod M = 3$$$
$$$(6 - 3) \mod M = 3 \mod M = 3$$$
$$$(7 - 4) \mod M = 3 \mod M = 3$$$
$$$(8 - 5) \mod M = 3 \mod M = 3$$$
$$$(9 - 6) \mod M = 3 \mod M = 3$$$
$$$(10 - 7) \mod M = 3 \mod M = 3$$$
$$$(11 - 8) \mod M = 3 \mod M = 3$$$
$$$(12 - 9) \mod M = 3 \mod M = 3$$$

As a result, $$$(0 + 998244345 + 998244345 + 998244345 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3) \mod 10^9 + 7$$$ $$$=$$$ $$$2994733059 \mod 10^9 + 7$$$ $$$=$$$ $$$994733045$$$

Informação

Provedor Codeforces

Código CF1601F