Score : $525$ points

问题陈述

存在一个简单的有向图 $G$，包含 $N$ 个顶点和 $N+M$ 条边。顶点编号从 $1$ 到 $N$，边编号从 $1$ 到 $N+M$。

边 $i$ $(1 \leq i \leq N)$ 从顶点 $i$ 指向顶点 $i+1$。（这里，顶点 $N+1$ 被视为顶点 $1$。）
边 $N+i$ $(1 \leq i \leq M)$ 从顶点 $X_i$ 指向顶点 $Y_i$。

高桥位于顶点 $1$。在每个顶点，他可以移动到从当前顶点有出边指向的任何顶点。

计算他恰好移动 $K$ 次的方式数量。

即，找出满足以下所有三个条件的整数序列 $(v_0, v_1, \dots, v_K)$ 的长度为 $K+1$ 的数量：

• 对于 $i = 0, 1, \dots, K$，有 $1 \leq v_i \leq N$。
• $v_0 = 1$。
• 对于 $i = 1, 2, \ldots, K$，存在一条从顶点 $v_{i-1}$ 指向顶点 $v_i$ 的有向边。

由于这个数字可能非常大，请打印出模 $998244353$ 的结果。

以上为大语言模型 kimi 翻译，仅供参考。

Problem Statement

There is a simple directed graph $G$ with $N$ vertices and $N+M$ edges. The vertices are numbered $1$ to $N$, and the edges are numbered $1$ to $N+M$.

Edge $i$ $(1 \leq i \leq N)$ goes from vertex $i$ to vertex $i+1$. (Here, vertex $N+1$ is considered as vertex $1$.)
Edge $N+i$ $(1 \leq i \leq M)$ goes from vertex $X_i$ to vertex $Y_i$.

Takahashi is at vertex $1$. At each vertex, he can move to any vertex to which there is an outgoing edge from the current vertex.

Compute the number of ways he can move exactly $K$ times.

That is, find the number of integer sequences $(v_0, v_1, \dots, v_K)$ of length $K+1$ satisfying all of the following three conditions:

• $1 \leq v_i \leq N$ for $i = 0, 1, \dots, K$.
• $v_0 = 1$.
• There is a directed edge from vertex $v_{i-1}$ to vertex $v_i$ for $i = 1, 2, \ldots, K$.

Since this number can be very large, print it modulo $998244353$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq 50$
• $1 \leq K \leq 2 \times 10^5$
• $1 \leq X_i, Y_i \leq N$, $X_i \neq Y_i$
• All of the $N+M$ directed edges are distinct.
• All input values are integers.

Input

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

$N$ $M$ $K$

$X_1$ $Y_1$

$X_2$ $Y_2$

$\vdots$

$X_M$ $Y_M$

Output

Print the count modulo $998244353$.

Sample Input 1

6 2 5
1 4
2 5

Sample Output 1

5

The above figure represents the graph $G$. There are five ways for Takahashi to move:

• Vertex $1 \to$ Vertex $2 \to$ Vertex $3 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6$
• Vertex $1 \to$ Vertex $2 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $2$
• Vertex $1 \to$ Vertex $2 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $4$
• Vertex $1 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $2$
• Vertex $1 \to$ Vertex $4 \to$ Vertex $5 \to$ Vertex $6 \to$ Vertex $1 \to$ Vertex $4$

Sample Input 2

10 0 200000

Sample Output 2

1

Sample Input 3

199 10 1326
122 39
142 49
164 119
197 127
188 145
69 80
6 120
24 160
18 154
185 27

Sample Output 3

451022766