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 = '# cook your dish here\nn,m=map(int,input().split())\nR=n\nC=m\na=[None]*n\nz=[0]*n\no=[1]*n\nfor i in range(m):\n    if i%2!=0:\n        z[i]=1\n        o[i]=0\nzi=[None]*n\noi=[None]*n\nfor i in range(n):\n    a[i]=input()\n    a[i]=list(a[i])\n    for j in range(m):\n        a[i][j]=int(a[i][j])\n    \n    if i%2==0:\n        zi[i]=z[:]\n        oi[i]=o[:]\n    else:\n        zi[i]=o[:]\n        oi[i]=z[:]\nforzi=0\nforoi=0\npsao = [[0 for x in range(C)] for y in range(R)]  \npsaz = [[0 for x in range(C)] for y in range(R)] \nnoioo=0\nnoizz=0\nnoio=0\nnoiz=0\nfor i in range(n):\n    noiz=0\n    noio=0\n    \n    for j in range(m):\n        \n        if i==0:\n            if a[i][j]!=oi[i][j]:\n                noio+=1\n            else:\n                noiz+=1\n            psao[i][j]=noio\n            psaz[i][j]=noiz\n        else:\n            if a[i][j]!=oi[i][j]:\n                noio+=1\n            else:\n                noiz+=1\n            psaz[i][j]=noiz+psaz[i-1][j]\n            psao[i][j]=noio+psao[i-1][j]\n    \n\n        \n    \n\n\nmmm=[]\nfor i in range(2,min(m,n)+1):\n    c=0\n    ap=[]\n    for j in range(n-i+1):\n        for k in range(m-i+1):\n            czz=0\n            czz1=0\n            czz2=0\n            coo=0\n            cor=0\n            czz+=(psaz[j+i-1][k+i-1])\n            coo+=(psao[j+i-1][k+i-1])\n\n            if k-1>=0:\n                czz-=(psaz[j+i-1][k-1])\n                coo-=(psao[j+i-1][k-1])\n            else:\n                czz+=0\n            if j-1>=0:\n                czz-=(psaz[j-1][k+i-1])\n                coo-=(psao[j-1][k+i-1])\n\n            else:\n                czz+=0\n            if j-1>=0 and k-1>=0:\n                czz+=(psaz[j-1][k-1])\n                coo+=(psao[j-1][k-1])\n            else:\n                czz+=0\n            \n            ap.append((min((abs(coo)),abs(czz))))\n    mmm.append(min(ap))\n\nq=int(input())\nlo=[int(pp) for pp in input().split()]\nmaxq=max(mmm)\nfor i in range(q):\n    if lo[i]>maxq:\n        print(min(n,m))\n    else:\n        j=len(mmm)-1\n        ind=0\n        while j>=0:\n            if lo[i]>=mmm[j]:\n                ind=j+2\n                break\n            j-=1\n        print(ind)'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 15029