Given a positive integer n, break it into the sum of at least two positive integers and maximize the product of those integers. Return the maximum product you can get.



For example, given n = 2, return 1 (2 = 1 + 1); given n = 10, return 36 (10 = 3 + 3 + 4).



Note: You may assume that n is not less than 2 and not larger than 58.


Credits:Special thanks to @jianchao.li.fighter for adding this problem and creating all test cases.

```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 integerBreak(self, n):\n         """\n         :type n: int\n         :rtype: int\n         """\n         if n < 4:\n             return n - 1\n \n         #  Proof.\n         #  1. Let n = a1 + a2 + ... + ak, product = a1 * a2 * ... * ak\n         #      - For each ai >= 4, we can always maximize the product by:\n         #        ai <= 2 * (ai - 2)\n         #      - For each aj >= 5, we can always maximize the product by:\n         #        aj <= 3 * (aj - 3)\n         #  \n         #     Conclusion 1:\n         #      - For n >= 4, the max of the product must be in the form of\n         #        3^a * 2^b, s.t. 3a + 2b = n\n         # \n         #  2. To maximize the product = 3^a * 2^b s.t. 3a + 2b = n\n         #      - For each b >= 3, we can always maximize the product by:\n         #        3^a * 2^b <= 3^(a+2) * 2^(b-3) s.t. 3(a+2) + 2(b-3) = n\n         # \n         #     Conclusion 2:\n         #      - For n >= 4, the max of the product must be in the form of\n         #        3^Q * 2^R, 0 <= R < 3 s.t. 3Q + 2R = n\n         #        i.e.\n         #          if n = 3Q + 0,   the max of the product = 3^Q * 2^0\n         #          if n = 3Q + 2,   the max of the product = 3^Q * 2^1\n         #          if n = 3Q + 2*2, the max of the product = 3^Q * 2^2\n \n         res = 0\n         if n % 3 == 0:            #  n = 3Q + 0, the max is 3^Q * 2^0\n             res = 3 ** (n // 3)\n         elif n % 3 == 2:          #  n = 3Q + 2, the max is 3^Q * 2^1\n             res = 3 ** (n // 3) * 2\n         else:                     #  n = 3Q + 4, the max is 3^Q * 2^2\n             res = 3 ** (n // 3 - 1) * 4\n         return res   \n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'integerBreak'
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: 5398