*This is the advanced version of the 
[Minimum and Maximum Product of k Elements](https://www.codewars.com/kata/minimum-and-maximum-product-of-k-elements/) kata.*

---

Given a list of **integers** and a positive integer `k` (> 0), find the minimum and maximum possible product of `k` elements taken from the list.

If you cannot take enough elements from the list, return `None`/`nil`.

## Examples

```python
numbers = [1, -2, -3, 4, 6, 7]

k = 1  ==>  -3, 7
k = 2  ==>  -21, 42    # -3*7, 6*7
k = 3  ==>  -126, 168  # -3*6*7, 4*6*7
k = 7  ==>  None       # there are only 6 elements in the list
```

Note: the test lists can contain up to 500 elements, so a naive approach will not work.

---

## My other katas

If you enjoyed this kata then please try [my other katas](https://www.codewars.com/collections/katas-created-by-anter69)! :-)

#### *Translations are welcome!*

```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 = 'import random, math\n\n# even if k is on the order of n, the average case is O(n)\n# worst case is O(n^2)\ndef find_min_max_product(arr, k):\n    if k < 1 or k > len(arr) or len(arr) == 0:\n        return None\n    \n    # get k largest values (by absolute value)\n    # and also construct a list of all the other values\n    k_vals, other_vals = partition_around_k(arr, k, True)\n        \n    largest_product = 1\n    smallest_positive_k_val = None\n    smallest_negative_k_val = None\n    # single loop through arr, so still O(n)\n    for num in k_vals:\n        # calculate the largest possible product by absolute value\n        largest_product *= num\n        # get the smallest positive and smallest negative values in k_vals\n        if num > 0 and (smallest_positive_k_val == None or num < smallest_positive_k_val):\n            smallest_positive_k_val = num\n        if num < 0 and (smallest_negative_k_val == None or num > smallest_negative_k_val):\n            smallest_negative_k_val = num\n    \n    # min and max are the same\n    if k == len(arr):\n        return largest_product, largest_product\n    \n    # if the largest product was positive: now find the largest negative product\n    # if the largest product was negative: now find the largest positive product\n    other_min_or_max = negate_product(arr, largest_product, smallest_negative_k_val, smallest_positive_k_val, other_vals)\n\n    # if there was no way to flip the sign, instead find the smallest product by absolute value\n    if other_min_or_max == None:\n        # this mirrors the logic above, except with the seek_largest param flipped\n        k_vals, other_vals = partition_around_k(arr, k, False)\n        smallest_product = 1\n        for num in k_vals:\n            smallest_product *= num\n        other_min_or_max = smallest_product\n\n    # always return the min followed by max\n    if largest_product > other_min_or_max:\n        return other_min_or_max, largest_product\n    else:\n        return largest_product, other_min_or_max\n\n# find the other min or max by flipping one of its factors from positive to negative or vice versa\ndef negate_product(arr, largest_product, smallest_negative_k_val, smallest_positive_k_val, other_vals):\n    other_min_or_max = None\n    need_min = largest_product > 0\n    negate_removed = None\n    positive_removed = None\n    if smallest_negative_k_val != None:\n        negate_removed = largest_product // smallest_negative_k_val\n    if smallest_positive_k_val != None:\n        positive_removed = largest_product // smallest_positive_k_val\n    \n    # single loop through arr, so still O(n)\n    for num in other_vals:\n        # calculate new possible product\n        if num >= 0 and negate_removed != None:\n            product = negate_removed * num\n        elif num <= 0 and positive_removed != None:\n            product = positive_removed * num\n        else:\n            continue\n        \n        # update the min or max\n        if other_min_or_max == None:\n            other_min_or_max = product\n            continue\n        is_product_better = product < other_min_or_max if need_min else product > other_min_or_max\n        if is_product_better:\n            other_min_or_max = product\n            \n    return other_min_or_max\n\n# find k-th largest element, by distance from zero\n# average case O(n), worst case is O(n^2)\ndef quick_select_absolute_k(arr, k, seek_largest):\n    # base cases\n    if len(arr) == 1:\n        return arr[0]\n    if k == 1:\n        return find_absolute_extreme(arr, seek_largest)\n    \n    pivot_index = pick_pivot_index(arr, 0, len(arr) - 1)\n    smaller = []\n    larger = []\n    same = []\n    for num in arr:\n        if abs(num) < abs(arr[pivot_index]):\n            smaller.append(num)\n        elif abs(num) > abs(arr[pivot_index]):\n            larger.append(num)\n        else:\n            if seek_largest:\n                larger.append(num)\n            else:\n                smaller.append(num)\n            same.append(num)\n    \n    # everything remaining is duplicates\n    if len(same) == len(arr):\n        return arr[0]\n    \n    preferred_array = larger if seek_largest else smaller\n    other_array = smaller if seek_largest else larger\n    if len(preferred_array) >= k:\n        return quick_select_absolute_k(preferred_array, k, seek_largest)\n    else:\n        return quick_select_absolute_k(other_array, k - len(preferred_array), seek_largest)\n\n# change to median-of-medians to improve worst case from O(n^2) to O(n)\ndef pick_pivot_index(arr, min, max):\n    return random.randint(min, max)\n    \ndef find_absolute_extreme(arr, seek_largest):\n    extreme = None\n    for num in arr:\n        if extreme == None:\n            extreme = num\n        elif seek_largest and abs(extreme) < abs(num):\n            extreme = num\n        elif not seek_largest and abs(extreme) > abs(num):\n            extreme = num\n    return extreme\n\ndef partition_around_k(arr, k, seek_largest):\n    k_val = quick_select_absolute_k(arr, k, seek_largest)\n    k_vals = []\n    k_val_copies = []\n    other_vals = []\n    # single loop through arr, so still O(n)\n    for num in arr:\n        if abs(num) > abs(k_val):\n            if seek_largest:\n                k_vals.append(num)\n            else:\n                other_vals.append(num)\n        elif abs(num) < abs(k_val):\n            if seek_largest:\n                other_vals.append(num)\n            else:\n                k_vals.append(num)\n        else:\n            k_val_copies.append(num)\n    \n    # handling for duplicates\n    remainder = k - len(k_vals)\n    for i in range(remainder):\n        k_vals.append(k_val_copies[i])\n    for i in range(remainder, len(k_val_copies)):\n        other_vals.append(k_val_copies[i])\n    \n    return k_vals, other_vals'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'find_min_max_product'
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: 9494