AT_arc202_d [ARC202D] King

There is a shogi (Japanese chess) board with $ H $ rows and $ W $ columns, and one king piece. The square at the $ i $ -th row from the top and $ j $ -th column from the left of the board is represented as $ (i, j) $ . Initially, the king is placed at $ (A, B) $ . You perform the following operation exactly $ T $ times. - Move the king to one of the eight adjacent squares of the current square. More precisely, if the king is at square $ (i, j) $ , move the king to one of $ (i+1,j+1), (i+1,j), (i+1,j-1), (i,j+1), (i,j-1), (i-1,j+1), (i-1,j), (i-1,j-1) $ . The king cannot leave the board. Find the number, modulo $ 998244353 $ , of operation sequences such that the king is at $ (C, D) $ after completing all $ T $ operations. Two operation sequences are considered different if and only if there exists an integer $ i $ $ (1 \leq i \leq T) $ such that the king's position after the $ i $ -th operation is different.

The input is given from Standard Input in the following format: > $ H $ $ W $ $ T $ $ A $ $ B $ $ C $ $ D $

Output the number, modulo $ 998244353 $ , of operation sequences such that the king is at $ (C, D) $ after completing all $ T $ operations.

### Sample Explanation 1 There are five operation sequences that satisfy the conditions in the problem statement. The king's movements corresponding to each operation sequence are as follows. - $ (2, 1) \to (1, 2) \to (2, 3) \to (3, 4) $ - $ (2, 1) \to (2, 2) \to (2, 3) \to (3, 4) $ - $ (2, 1) \to (2, 2) \to (3, 3) \to (3, 4) $ - $ (2, 1) \to (3, 2) \to (2, 3) \to (3, 4) $ - $ (2, 1) \to (3, 2) \to (3, 3) \to (3, 4) $ ### Constraints - $ 2 \leq H \leq 3 \times 10^5 $ - $ 2 \leq W \leq 3 \times 10^5 $ - $ 1 \leq T \leq 3 \times 10^5 $ - $ 1 \leq A \leq H $ - $ 1 \leq B \leq W $ - $ 1 \leq C \leq H $ - $ 1 \leq D \leq W $ - All input values are integers.