Given a rectangle of size n x m, find the minimum number of integer-sided squares that tile the rectangle.
 
Example 1:

Input: n = 2, m = 3
Output: 3
Explanation: 3 squares are necessary to cover the rectangle.
2 (squares of 1x1)
1 (square of 2x2)
Example 2:

Input: n = 5, m = 8
Output: 5

Example 3:

Input: n = 11, m = 13
Output: 6

 
Constraints:

1 <= n <= 13
1 <= m <= 13

```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

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'class Solution:\n    def tilingRectangle(self, n: int, m: int) -> int:\n        # from https://leetcode.com/problems/tiling-a-rectangle-with-the-fewest-squares/discuss/414716/Python-two-solutions%3A-backtracking-and-A*-search\n        # backtracking\n        # fill all spaces with all possible sizes one by one\n        # like stacking in storage, put one box at a time\n        # that covers the floor, i.e. no gap underneath, e.g. box-1 in example 3\n        # (examples call it skyline filling)\n        # algo:\n        # maintain an array of heights with unit widths\n        # each iteration, fill the min height space\n        # not greedily, but try all sizes\n        # pick larger size first to slightly improve search\n        # so that smaller size resulting in more boxes than a found solution\n        # can be skipped\n        \n        self.best = m * n # max number, also guaranteed to fill\n        \n        def dfs(heights, count):\n            if min(heights) == n: # filled out all unit columns\n                self.best = min(self.best, count)\n                return \n            if count >= self.best: # a solution with count already exists, not point searching\n                return \n            \n            # find minimum height spacce to fill \n            min_height = min(heights)\n            idx = heights.index(min_height)\n            right_end = idx + 1\n            while right_end < m and heights[right_end] == min_height:\n                right_end += 1\n            max_possible_box = min(right_end-idx, n - min_height)\n            \n            # now fill with boxes and backtrack\n            for box_size in range(max_possible_box, 0, -1):\n                new_heights = heights[:]\n                for i in range(box_size):\n                    new_heights[idx+i] += box_size\n                dfs(new_heights, count + 1)\n            \n        dfs([0] * m, 0)\n        return self.best'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'tilingRectangle'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 7324