Chef loves to play chess, so he bought a new chessboard with width M$M$ and height N$N$ recently.
Chef considers a chessboard correct if its width (number of columns) is equal to its height (number of rows) and each cell has no side-adjacent cell of the same color (this is the so-called "chess order" which you can see in real-world chessboards). Chef's chessboard does not have to be a correct chessboard (in particular, it may have N≠M$N \neq M$).
A sub-board of Chef's chessboard is a rectangular piece of this board with an arbitrarily chosen top left and bottom right cell (possibly equal to the original chessboard). Every sub-board is also a chessboard.
Chef can invert some cells; inverting a cell means changing its color from white to black or from black to white. After inverting those cells, he wants to cut the maximum correct sub-board out of the original chessboard.
Chef has not yet decided how many cells he would like to invert. Now he wonders about the answers to Q$Q$ question. In the i$i$-th question (1≤i≤Q$1 \le i \le Q$), he is allowed to invert at most ci$c_i$ cells (possibly zero); he would like to know the side length of the largest possible correct sub-board of his chessboard. Help Chef answer these questions.

-----Input-----
- The first line of the input contains two space-separated integers N$N$ and M$M$.
- N$N$ lines follow. For each valid i$i$, the i$i$-th of these lines contains a string with length M$M$ describing the i$i$-th row of Chef's chessboard. Each character of this string is either '0', representing a black cell, or '1', representing a white cell.
- The next line contains a single integer Q$Q$.
- The last line contains Q$Q$ space-separated integers c1,c2,…,cQ$c_1, c_2, \dots, c_Q$.

-----Output-----
For each question, print a single line containing one integer — the maximum size of a correct sub-board.

-----Constraints-----
- 1≤N,M≤200$1 \le N, M \le 200$
- 1≤Q≤105$1 \le Q \le 10^5$
- 0≤ci≤109$0 \le c_i \le 10^9$ for each valid i$i$

-----Subtasks-----
Subtask #1 (20 points):
- 1≤N,M≤20$1 \le N, M \le 20$
- 1≤Q≤100$1 \le Q \le 100$
Subtask #2 (30 points): 1≤N,M≤20$1 \le N, M \le 20$
Subtask #3 (50 points): original constraints

-----Example Input-----
8 8
00101010
00010101
10101010
01010101
10101010
01010101
10101010
01010101
4
1 2 0 1001

-----Example Output-----
7
8
6
8

-----Explanation-----
If we don't change the board, the best answer here is the 6x6 bottom right sub-board. We can invert cells (2,2)$(2, 2)$ and (1,1)$(1, 1)$ to get a better answer.

```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 = 'n,m=map(int,input().split())\r\nchess=[]\r\nfor _ in range(n):\r\n    chess.append([int(x) for x in list(input())])\r\nchessO=[]\r\nchessL=[]\r\nfixO=[]\r\nfixL=[]\r\nfor _ in range(n):\r\n    p=[0]*m\r\n    fixL.append(p)\r\n    q=[0]*m\r\n    fixO.append(q)\r\n    tm=[0]*m\r\n    chessO.append(tm)\r\n    to=[0]*m\r\n    chessL.append(to)\r\n\r\nfor i in range(n):\r\n    for j in range(m):\r\n        chessO[i][j]=((i+j)%2)^chess[i][j]\r\n        chessL[i][j]=((i+j+1)%2)^chess[i][j]\r\n        fixO[i][j]=chessO[i][j]\r\n        fixL[i][j]=chessL[i][j]\r\n\r\nfor i in range(n):\r\n    for j in range(1,m):\r\n        fixO[i][j]=fixO[i][j-1]+chessO[i][j]\r\n        fixL[i][j]=fixL[i][j-1]+chessL[i][j]\r\n\r\nans=[-1]*min(m,n)\r\nfor i in range(n):\r\n    for j in range(m):\r\n        for k in range(min(n-i,m-j)):\r\n            errorO,errorL=0,0\r\n            for l in range(i,i+k+1):\r\n                errorO+=fixO[l][j+k]-fixO[l][j]+chessO[l][j]\r\n                errorL+=fixL[l][j+k]-fixL[l][j]+chessL[l][j]\r\n            if ans[k]==-1 or ans[k]>min(errorO,errorL): ans[k]=min(errorO,errorL)\r\nfinal=[0]*50000\r\nmaxo=9999999999\r\nfor i in range(min(m,n)-1,-1,-1):\r\n    if ans[i]!=-1 and ans[i]<maxo:\r\n        maxo=ans[i]\r\n        final[maxo]=i\r\ntmp=final[0]\r\nfor i in range(1,50000):\r\n    if  final[i]>tmp:\r\n        tmp=final[i]\r\n    final[i]=tmp\r\ninput()\r\nq=[int(x) for x in input().split()]\r\n\r\nfor x in q:\r\n    if x>50000-2 : print(final[45000]+1)\r\n    else: print(final[x]+1)\r\n    \r\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 413062