We are given a certain number ```n``` and we do the product partitions of it.
```[59, 3, 2, 2, 2]``` is a product partition of ```1416``` because:
```
59 * 3 * 2 * 2 * 2 = 1416
```
We form a score, ```sc``` for each partition in the following way:
- if ```d1, d2, ...., dk``` are the prime factors of ```n```, and ```f1, f2, ...., fk```, the corresponding frequencies for each factor, we calculate:



Suposse that we have that ```n = 1416``` 
The product partitions of this number with a corresponding special score are as follows:
```
Product Partition           Score(sc)
[59, 3, 2, 2, 2]             350    # equals to: (59^1 + 3^1 + 2^3) * 5
[177, 2, 2, 2]               740    # equals to: (177^1 + 2^3) * 4
[118, 3, 2, 2]               500
[59, 6, 2, 2]                276
[354, 2, 2]                 1074
[59, 4, 3, 2]                272
[236, 3, 2]                  723
[177, 4, 2]                  549
[118, 6, 2]                  378
[59, 12, 2]                  219
[708, 2]                    1420  <---- maximum value
[118, 4, 3]                  375
[59, 8, 3]                   210
[472, 3]                     950
[59, 6, 4]                   207
[354, 4]                     716
[236, 6]                     484
[177, 8]                     370
[118, 12]                    260
[59, 24]                     166  <---- minimum value
```
So we need a function that may give us the product partition with maximum or minimum score.

The function ```find_spec_prod_part()``` will receive two arguments:

- an integer ```n, n > 0```
- a command as a string, one of the following ones: ```'max' or 'min'```

The function should output a list with two elements: the found product partition (as a list sorted in descendin order) with its corresponding score.
```
find_spec_prod_part(n, com) ---> [prod_partition, score]
```
Let'see some cases:
```python
find_spec_prod_part(1416, 'max') == [[708, 2], 1420]

find_spec_prod_part(1416, 'min') == [[59, 24], 166]
```

The function should reject prime numbers:
```python
find_spec_prod_part(10007 , 'max') == "It is a prime number"
```
Enjoy it!

Hint: In this kata, optimization is one of the purposes or tags. The algorithm to produce the product partition is a key factor in terms of speed. Your code will be tested for an ```n``` value up to ```500000```.

```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 collections import Counter\nfrom itertools import permutations\nfrom itertools import chain\nimport numpy as np\n\n\ndef prime_factors(n):\n    factors = []\n    while n % 2 == 0:\n        n = n // 2\n        factors.append(2)\n        \n    for k in range(3, n+1, 2):\n        while n % k == 0:\n            n = n // k\n            factors.append(k)\n        if n == 1:\n            break\n    return factors\n\n   \ndef get_score(factors):\n    factor_counts = Counter(factors)\n    return sum(f**factor_counts[f] for f in factor_counts)*sum(c for c in factor_counts.values())\n\n\ndef int_partitions(m, memo = {}):\n    if m in memo:\n        return memo[m]\n    all_partitions = [[m]]\n    \n    for i in range(1, m):\n        for p in int_partitions(m-i, memo):\n            all_partitions.append([i] + p)\n    \n    memo[m] = all_partitions\n    return all_partitions\n\n\ndef make_partitions(factors):\n    partitions = int_partitions(len(factors))\n    part_perm = []\n    \n    for p in partitions:\n        part_perm.append(set(list(permutations(p, len(p)))))\n\n    part_perm = set(list(chain.from_iterable(part_perm)))\n    all_new_factors = []\n\n    for inds in part_perm:        \n        new_factors = []\n        j_start = 0\n\n        for i in inds:\n            j_end = j_start + i\n            new_factors.append(np.product(factors[j_start:j_end]))\n            j_start = j_end\n        \n        if len(new_factors) > 1:\n            all_new_factors.append(new_factors) \n    \n    return all_new_factors\n\n\ndef find_spec_prod_part(n, com):\n    factors = prime_factors(n)   \n    \n    if len(factors) == 1:\n        return "It is a prime number"\n    \n    all_factors = make_partitions(factors)\n    scores = [get_score(x) for x in all_factors]\n    \n    if com == "max":\n        opt_id = np.argmax(scores)\n    else:\n        opt_id = np.argmin(scores)\n    \n    return [sorted(all_factors[opt_id], reverse=True), scores[opt_id]] '
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'find_spec_prod_part'
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: