Codeforces Round 493 (Div. 1) |
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Finished |

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combinatorics

divide and conquer

trees

*2900

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D. Cycles in product

time limit per test

7 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputConsider a tree (that is, an undirected connected graph without loops) $$$T_1$$$ and a tree $$$T_2$$$. Let's define their cartesian product $$$T_1 \times T_2$$$ in a following way.

Let $$$V$$$ be the set of vertices in $$$T_1$$$ and $$$U$$$ be the set of vertices in $$$T_2$$$.

Then the set of vertices of graph $$$T_1 \times T_2$$$ is $$$V \times U$$$, that is, a set of ordered pairs of vertices, where the first vertex in pair is from $$$V$$$ and the second — from $$$U$$$.

Let's draw the following edges:

- Between $$$(v, u_1)$$$ and $$$(v, u_2)$$$ there is an undirected edge, if $$$u_1$$$ and $$$u_2$$$ are adjacent in $$$U$$$.
- Similarly, between $$$(v_1, u)$$$ and $$$(v_2, u)$$$ there is an undirected edge, if $$$v_1$$$ and $$$v_2$$$ are adjacent in $$$V$$$.

Please see the notes section for the pictures of products of trees in the sample tests.

Let's examine the graph $$$T_1 \times T_2$$$. How much cycles (not necessarily simple) of length $$$k$$$ it contains? Since this number can be very large, print it modulo $$$998244353$$$.

The sequence of vertices $$$w_1$$$, $$$w_2$$$, ..., $$$w_k$$$, where $$$w_i \in V \times U$$$ called cycle, if any neighboring vertices are adjacent and $$$w_1$$$ is adjacent to $$$w_k$$$. Cycles that differ only by the cyclic shift or direction of traversal are still considered different.

Input

First line of input contains three integers — $$$n_1$$$, $$$n_2$$$ and $$$k$$$ ($$$2 \le n_1, n_2 \le 4000$$$, $$$2 \le k \le 75$$$) — number of vertices in the first tree, number of vertices in the second tree and the cycle length respectively.

Then follow $$$n_1 - 1$$$ lines describing the first tree. Each of this lines contains two integers — $$$v_i, u_i$$$ ($$$1 \le v_i, u_i \le n_1$$$), which define edges of the first tree.

Then follow $$$n_2 - 1$$$ lines, which describe the second tree in the same format.

It is guaranteed, that given graphs are trees.

Output

Print one integer — number of cycles modulo $$$998244353$$$.

Examples

Input

2 2 2

1 2

1 2

Output

8

Input

2 2 4

1 2

1 2

Output

32

Input

2 3 4

1 2

1 2

1 3

Output

70

Input

4 2 2

1 2

1 3

1 4

1 2

Output

20

Note

The following three pictures illustrate graph, which are products of the trees from sample tests.

In the first example, the list of cycles of length $$$2$$$ is as follows:

- «AB», «BA»
- «BC», «CB»
- «AD», «DA»
- «CD», «DC»

Codeforces (c) Copyright 2010-2023 Mike Mirzayanov

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