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Ex - Dice Sum 2

Score : $600$ points

Problem Statement

The six-sided dice speciality shop "Saikoroya" sells $N$ dice. The $i$-th die (singular of dice) has $A_{i,1},A_{i,2},\ldots,A_{i,6}$ written on its each side, and has a price of $C_i$.

Takahashi is going to choose exactly $K$ of them and buy them.

Currently, "Saikoroya" is conducting a promotion: Takahashi may roll each of the purchased dice once and claim money whose amount is equal to the square of the sum of the numbers shown by the dice. Here, each die shows one of the six numbers uniformly at random and independently.

Maximize the expected value of (the amount of money he claims) - (the sum of money he pays for the purchased $K$ dice) by properly choosing $K$ dice to buy. Print the maximized expected value modulo $998244353$.

Constraints

• $1 \leq N \leq 1000$
• $1 \leq K \leq N$
• $1 \leq C_i \leq 10^5$
• $1 \leq A_{i,j} \leq 10^5$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$

$C_1$ $C_2$ $\ldots$ $C_N$

$A_{1,1}$ $A_{1,2}$ $\ldots$ $A_{1,6}$

$\vdots$

$A_{N,1}$ $A_{N,2}$ $\ldots$ $A_{N,6}$

Output

Print the answer.

3 2
1 2 3
1 1 1 1 1 1
2 2 2 2 2 2
3 3 3 3 3 3

20

If he buys the $2$-nd and $3$-rd dice, the expected value of (the amount of money he claims) - (the sum of money he pays for the purchased $K$ dice) equals $(2 + 3)^2 - (2 + 3) = 20$, which is the maximum expected value.

10 5
2 5 6 5 2 1 7 9 7 2
5 5 2 4 7 6
2 2 8 7 7 9
8 1 9 6 10 8
8 6 10 3 3 9
1 10 5 8 1 10
7 8 4 8 6 5
1 10 2 5 1 7
7 4 1 4 5 4
5 10 1 5 1 2
5 1 2 3 6 2

1014