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 = "from functools import lru_cache\n\nclass Solution:\n    def tilingRectangle(self, n: int, m: int) -> int:\n        @lru_cache(maxsize=None)\n        def dfs(x, y):\n            if x == y:\n                return 1\n            if x == 1:\n                return y\n            if y == 1:\n                return x\n\n            # Max possible result full of squares of side 1\n            result = x * y\n\n            # Splitting x into two parts\n            for i in range(1, (x // 2) + 1):\n                left = dfs(i, y)\n                right = dfs(x-i, y)\n                result = min(result, left + right)\n\n            # Splitting y into two parts\n            for k in range(1, (y // 2) + 1):\n                bottom = dfs(x, k)\n                top = dfs(x, y-k)\n                result = min(result, bottom + top)\n\n            # A central square with 4 other rectangles bounding it\n            for size in range(1, min(x, y)):\n                for i in range(1, x-size):\n                    for k in range(1, y-size):\n                        partition1 = dfs(i+size, k)\n                        partition2 = dfs(x-size-i, k+size)\n                        partition3 = dfs(x-i, y-size-k)\n                        partition4 = dfs(i, y-k)\n                        partition5 = 1 # A square of side 'size'. dfs(size, size) = 1\n\n                        curr_result = partition1 + partition2 + partition3 + partition4 + partition5\n                        result = min(result, curr_result)\n\n            return result\n\n        return dfs(n, m)\n"
_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: 8536