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 = '# -*- coding: utf-8 -*-\r\n\r\nimport math\r\nimport collections\r\nimport bisect\r\nimport heapq\r\nimport time\r\nimport random\r\nimport itertools\r\nimport sys\r\n\r\n"""\r\ncreated by shhuan at 2018/10/30 19:45\r\n\r\n"""\r\n\r\n\r\nN, M = map(int, input().split())\r\n\r\nA = []\r\n\r\nfor i in range(N):\r\n    A.append([int(x) for x in input()])\r\n\r\n# for row in A:\r\n#     print(row)\r\n\r\n\r\n#f(r, c, color, width) = f(r+1, c+1, color, width-1) + mismatches_in_edge\r\n\r\nmxMN, mnMN = max(M, N), min(M, N)\r\nans = [float(\'inf\') for _ in range(mxMN + 1)]\r\nf = [[[[0 for _ in range(mxMN + 1)] for color in range(2)] for c in range(M+1)] for r in range(N+1)]\r\n\r\nfor width in range(1, min(N, N)+1):\r\n    for color in [0, 1]:\r\n        for sr in range(N-width+1):\r\n            for sc in range(N-width+1):\r\n                steps = f[sr+1][sc+1][color][width-1]\r\n\r\n                cc = color\r\n                for c in range(sc, sc+width):\r\n                    steps += 0 if A[sr][c] == cc else 1\r\n                    cc ^= 1\r\n                cc = color ^ 1\r\n                for r in range(sr+1, sr+width):\r\n                    steps += 0 if A[r][sc] == cc else 1\r\n                    cc ^= 1\r\n\r\n                f[sr][sc][color][width] = steps\r\n                ans[width] = min(ans[width], steps)\r\n\r\nqc = input()\r\nqs = [int(x) for x in input().split()]\r\n\r\n# print()\r\n# print(ans)\r\nfor q in qs:\r\n    print(bisect.bisect_right(ans, q) - 1)\r\n\r\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

```

Measured memory_bytes: 47668