题目描述

— This is not playing but duty as allies of justice, Nii-chan!
— Not allies but justice itself, Onii-chan!

With hands joined, go everywhere at a speed faster than our thoughts! This time, the Fire Sisters — Karen and Tsukihi — is heading for somewhere they've never reached — water-surrounded islands!

There are three clusters of islands, conveniently coloured red, blue and purple. The clusters consist of $a$, $b$ and $c$ distinct islands respectively.

Bridges have been built between some (possibly all or none) of the islands. A bridge bidirectionally connects two different islands and has length 1. For any two islands of the same colour, either they shouldn't be reached from each other through bridges, or the shortest distance between them is at least 3, apparently in order to prevent oddities from spreading quickly inside a cluster.

The Fire Sisters are ready for the unknown, but they'd also like to test your courage. And you're here to figure out the number of different ways to build all bridges under the constraints, and give the answer modulo $998\ 244\ 353$. Two ways are considered different if a pair of islands exist, such that there's a bridge between them in one of them, but not in the other.

题目中文描述

有$3$个群岛$A、B、C$，分别有$a、b、c$个小岛。在其中建桥，要求 一个群岛内的任意两个小岛至少需要跨过3座桥才能到达。求建 桥方案数，答案对$998244353$取模。 数据范围 ：$a,b,c≤5000$

输入描述

The first and only line of input contains three space-separated integers $a$, $b$ and $c$ ($1 \le a, b, c \le 5\ 000$) — the number of islands in the red, blue and purple clusters, respectively.

输出描述

Output one line containing an integer — the number of different ways to build bridges, modulo $998\ 244\ 353$.

样例

1 1 1

8

说明 1

In the first example, there are 3 bridges that can possibly be built, and no setup of bridges violates the restrictions. Thus the answer is $2^3 = 8$.

1 2 2

63

1 3 5

3264

6 2 9

813023575

说明

In the second example, the upper two structures in the figure below are instances of valid ones, while the lower two are invalid due to the blue and purple clusters, respectively.