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\nimport numpy as np\n\nclass Solution:\n    def __init__(self):\n        self.cache={}\n        self.res=np.inf\n        \n    def tilingRectangle(self, n: int, m: int) -> int:\n        def tilingRectangleCalc(state, cnt):\n            if cnt>self.res:\n                return\n            state=list(state)\n            minHeight=min(state)\n            if minHeight==n:\n                self.res=min(self.res,cnt)\n                return\n            minIdx=state.index(minHeight)\n            maxL=1\n            while minIdx+maxL-1<=m-1 and state[minIdx+maxL-1]==minHeight:\n                maxL+=1\n            for l in range(maxL-1,0,-1):\n                if minHeight+l>n:\n                    continue\n                s=tuple(state[:minIdx]+[minHeight+l]*l+state[minIdx+l:])\n                if s in self.cache and self.cache[s]<cnt+1:\n                    continue\n                self.cache[s]=cnt+1\n                tilingRectangleCalc(s,cnt+1)\n            return\n        if m>n:\n            n,m=m,n\n        tilingRectangleCalc(tuple([0]*m),0)   \n        return self.res\n\n\n# class Solution:\n#     def tilingRectangle(self, n: int, m: int) -> int:\n#         @lru_cache(None)\n#         def tilingRectangleCalc(state):\n#             state=list(state)\n#             minHeight=min(state)\n#             if minHeight==n:\n#                 return 0\n#             minIdx=state.index(minHeight)\n#             res=np.inf\n#             for l in range(1,n-minHeight+1):\n#                 if minIdx+l-1>m-1 or state[minIdx+l-1]>minHeight:\n#                     break\n#                 res=min(res,tilingRectangleCalc(tuple(state[:minIdx]+\\\\\n#                                                       [minHeight+l]*l+state[minIdx+l:])))\n#             return 1+res\n#         if m>n:\n#             n,m=m,n\n#         return tilingRectangleCalc(tuple([0]*m))     \n\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: 8494