p03833 AtCoder Regular Contest 067 - Yakiniku Restaurants
There are N barbecue restaurants along a street. The restaurants are numbered 1 through N from west to east, and the distance between restaurant i and restaurant i+1 is A_i.

Joisino has M tickets, numbered 1 through M. Every barbecue restaurant offers barbecue meals in exchange for these tickets. Restaurant i offers a meal of deliciousness B_{i,j} in exchange for ticket j. Each ticket can only be used once, but any number of tickets can be used at a restaurant.

Joisino wants to have M barbecue meals by starting from a restaurant of her choice, then repeatedly traveling to another barbecue restaurant and using unused tickets at the restaurant at her current location. Her eventual happiness is calculated by the following formula: "(The total deliciousness of the meals eaten) - (The total distance traveled)". Find her maximum possible eventual happiness.

Constraints

* All input values are integers.
* 2≤N≤5×10^3
* 1≤M≤200
* 1≤A_i≤10^9
* 1≤B_{i,j}≤10^9

Input

The input is given from Standard Input in the following format:


N M
A_1 A_2 ... A_{N-1}
B_{1,1} B_{1,2} ... B_{1,M}
B_{2,1} B_{2,2} ... B_{2,M}
:
B_{N,1} B_{N,2} ... B_{N,M}


Output

Print Joisino's maximum possible eventual happiness.

Examples

Input

3 4
1 4
2 2 5 1
1 3 3 2
2 2 5 1


Output

11


Input

5 3
1 2 3 4
10 1 1
1 1 1
1 10 1
1 1 1
1 1 10


Output

20
```python
import sys
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines

import itertools

INF=10**10

N,M = map(int,readline().split())
A = [0,0]+list(map(int,readline().split()))
B = [[INF]*M]+[list(map(int,line.split())) for line in readlines()]

for i in range(1,N+1):
    A[i] += A[i-1]

answer = 0
INF = 10**12
diff = [INF]+[0]*N
left = [list(range(-1,N)) for m in range(M)]
arr = [INF,2,1,2]
for i in range(1,N+1):
    for m in range(M):
        p=i; prev_val=0
        q=left[m][p]
        x = B[i][m]
        while prev_val<=x:
            diff[p] += x-prev_val
            diff[q] -= x-prev_val
            p=q; q=left[m][p]
            prev_val=B[p][m]
        left[m][i]=p
    # 出発地点ごとの収入
    ticket_sum = list(itertools.accumulate(diff[::-1]))[::-1]
    for start in range(1,i+1):
        happy = ticket_sum[start] + A[start]-A[i]
        if answer < happy:
            answer = happy

print(answer)
```