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.

-----Sample Input-----
3 4
1 4
2 2 5 1
1 3 3 2
2 2 5 1

-----Sample Output-----
11

The eventual happiness can be maximized by the following strategy: start from restaurant 1 and use tickets 1 and 3, then move to restaurant 2 and use tickets 2 and 4.

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

ns = {"__name__": "__main__"}
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'import sys\nreadline = sys.stdin.readline\n\nN, M = map(int, readline().split())\nA = [0] + list(map(int, readline().split()))\nfor i in range(1, N):\n    A[i] += A[i-1]\n\nINF = 10**9+7\nB = [list(map(int, readline().split())) for _ in range(N)] + [[INF]*M] + [[0]*M]\nright = [[0]*M for _ in range(N)]\n\nfor m in range(M):\n    L = list(range(N))\n    L.sort(key = lambda x: B[x][m])\n    RR = list(range(1, N+3))\n    LL = list(range(-1, N+1))\n    for vn in L:\n        vn += 1\n        right[vn-1][m] = RR[vn]-1\n        RR[LL[vn]] = RR[vn]\n        LL[RR[vn]] = LL[vn]\n\ndiff = [0]*(N+1)\nfor m in range(M):\n    diff[0] += B[0][m]\n    pre = B[0][m]\n    for i in range(1, N):\n        if B[i][m] > pre:\n            diff[i] += B[i][m]-pre\n            pre = B[i][m]\n\noffset = 0\nans = 0\nfor l in range(N):\n    offset += diff[l]\n    for m in range(M):\n        if B[l][m] < B[l-1][m]:\n            oldr = right[l-1][m]\n            diff[oldr] -= B[oldr][m] - B[l-1][m]\n            offset += B[l][m] - B[l-1][m]\n            cnt = l\n            while cnt < N and right[cnt][m] <= oldr:\n                rcm = right[cnt][m]\n                diff[rcm] += B[rcm][m] - B[cnt][m]\n                cnt = rcm\n    ans = max(ans, offset)\n    roff = offset\n    for r in range(l+1, N):\n        roff += diff[r]\n        ans = max(ans, roff - (A[r]-A[l]))\n        #print(l, r, roff - (A[r] - A[l]))\n\nprint(ans)'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 14603