CF1792F1.Graph Coloring (easy version)

The only difference between the easy and the hard version is the constraint on $n$.

You are given an undirected complete graph on $n$ vertices. A complete graph is a graph where each pair of vertices is connected by an edge. You have to paint the edges of the graph into two colors, red and blue (each edge will have one color).

A set of vertices $S$ is red-connected if, for every pair of vertices $(v_1, v_2)$ such that $v_1 \in S$ and $v_2 \in S$, there exists a path from $v_1$ to $v_2$ that goes only through red edges and vertices from $S$. Similarly, a set of vertices $S$ is blue-connected if, for every pair of vertices $(v_1, v_2)$ such that $v_1 \in S$ and $v_2 \in S$, there exists a path from $v_1$ to $v_2$ that goes only through blue edges and vertices from $S$.

You have to paint the graph in such a way that:

• there is at least one red edge;
• there is at least one blue edge;
• for each set of vertices $S$ such that $|S| \ge 2$, $S$ is either red-connected or blue-connected, but not both.

Calculate the number of ways to paint the graph, and print it modulo $998244353$.

简单版本与困难版本的唯一区别在于 $n$ 的约束条件。

给你一个包含 $n$ 个顶点的无向完全图。完全图是指图中每一对顶点之间都有一条边相连。你需要将图中的所有边染成两种颜色之一：红色或蓝色（每条边恰好染一种颜色）。

若对顶点集合 $S$，任意两个顶点 $v_1, v_2 \in S$ 均存在一条仅由红色边及 $S$ 中的顶点构成的路径，则称 $S$ 是红连通的。类似地，若对任意 $v_1, v_2 \in S$，均存在一条仅由蓝色边及 $S$ 中的顶点构成的路径，则称 $S$ 是蓝连通的。

你需要对图进行染色，使得满足以下条件：

• 至少存在一条红色边；
• 至少存在一条蓝色边；
• 对每个满足 $|S| \ge 2$ 的顶点集合 $S$，$S$ 要么是红连通的，要么是蓝连通的，但不能两者同时成立。

计算满足上述条件的染色方案数，并将结果对 $998244353$ 取模后输出。

The first (and only) line contains one integer $n$ ($3 \le n \le 5 \cdot 10^3$).

第一行（也是唯一一行）包含一个整数 $n$（$3 \le n \le 5 \cdot 10^3$）。

Print one integer — the number of ways to paint the graph, taken modulo $998244353$.

输出一个整数——染色该图的方案数，对 $998244353$ 取模。