Coffee Vending Machine Problems [Part 1]

You have a vending machine, but it can not give the change back. You decide to implement this functionality. First of all, you need to know the minimum number of coins for this operation (i'm sure you don't want to return 100 pennys instead of 1$ coin).
So, find an optimal number of coins required, if you have unlimited set of coins with given denominations.

Assume all inputs are valid positive integers, and every set of coin denominations has len 4 for simplicity;

Examples:

optimal_number_of_coins(1, [1, 2, 5, 10]) 
(1 penny) so returns 1  
optimal_number_of_coins(5, [1, 2, 5, 10])
(5) so returns 1
optimal_number_of_coins(6, [1, 3, 5, 10])
(3+3 or 5+1) = 6 so returns 2
optimal_number_of_coins(10, [1, 2, 5, 10]) 
(10) so returns 1
optimal_number_of_coins(12, [1, 3, 5, 10])
(10+1+1) = 12 so returns 3
optimal_number_of_coins(53, [1, 2, 5, 25])
(25+25+2+1) = 53 so returns 4
optimal_number_of_coins(7, [1, 1, 1, 25])
(1+1+1+1+1+1+1) = 7 so returns 7
etc..

Have fun =)

```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 = 'def optimal_number_of_coins(n,A):\n    D={}\n    for i in range(1,n+1):D[i]=min(-~j+(i+~j*x and D[i+~j*x])for x in A for j in range(i//x))\n    return D[n]'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'optimal_number_of_coins'
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: 5932