I. Imitating the Key Treetime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputPak Chanek has a tree called the key tree. This tree consists of N vertices and N-1 edges. The edges of the tree are numbered from 1 to N-1 with edge i connecting vertices U_i and V_i. Initially, each edge of the key tree does not have a weight.Formally, a path with length k in a graph is a sequence [v_1, e_1, v_2, e_2, v_3, e_3, \ldots, v_k, e_k, v_{k+1}] such that:   For each i, v_i is a vertex and e_i is an edge.  For each i, e_i connects vertices v_i and v_{i+1}. A circuit is a path that starts and ends on the same vertex.A path in a graph is said to be simple if and only if the path does not use the same edge more than once. Note that a simple path can use the same vertex more than once.The cost of a simple path in a weighted graph is defined as the maximum weight of all edges it traverses.Count the number of distinct undirected weighted graphs that satisfy the following conditions:   The graph has N vertices and 2N-2 edges.  For each pair of different vertices (x, y), there exists a simple circuit that goes through vertices x and y in the graph.  The weight of each edge in the graph is an integer between 1 and 2N-2 inclusive. Each edge has distinct weights.  The graph is formed in a way such that there is a way to assign a weight W_i to each edge i in the key tree that satisfies the following conditions:   For each pair of edges (i, j), if i<j, then W_i<W_j.  For each pair of different vertex indices (x, y), the cost of the only simple path from vertex x to y in the key tree is equal to the minimum cost of a simple circuit that goes through vertices x and y in the graph.   Note that the graph is allowed to have multi-edges, but is not allowed to have self-loops. Print the answer modulo 998\,244\,353.Two graphs are considered distinct if and only if there exists a triple (a, b, c) such that there exists an edge that connects vertices a and b with weight c in one graph, but not in the other.InputThe first line contains a single integer N (2 \le N \le 10^5) — the number of vertices in the key tree.The i-th of the next N-1 lines contains two integers U_i and V_i (1 \le U_i, V_i \le N) — an edge connecting vertices U_i and V_i. The graph in the input is a tree.OutputAn integer representing the number of distinct undirected weighted graphs that satisfy the conditions of the problem modulo 998\,244\,353.ExampleInput
4
3 2
1 3
4 3
Output
540
NoteThe following is an example of a graph that satisfies.The following is an assignment of edge weights in the key tree that corresponds to the graph above.As an example, consider a pair of vertex indices (1, 4).   The circuit in the graph for this pair of vertices is 3 \xrightarrow{2} 2 \xrightarrow{4} 4 \xrightarrow{6} 2 \xrightarrow{1} 1 \xrightarrow{5} 3 with a cost of 6.  The path in the key tree for this pair of vertices is 1 \xrightarrow{5} 3 \xrightarrow{6} 4 with a cost of 6. 
4
3 2
1 3
4 3

540 
['combinatorics', 'dsu', 'trees', '*2800']