An expression is formed by taking the digits 1 to 9 in numerical order and then inserting into each gap between the numbers either a plus sign or a minus sign or neither.

Your task is to write a method which takes one parameter and returns the **smallest possible number** of plus and minus signs necessary to form such an expression which equals the input.

**Note:** All digits from 1-9 must be used exactly once.

If there is no possible expression that evaluates to the input, then return `null/nil/None`.

~~~if:haskell
`eval :: String -> Int` is available in `Preloaded` for your convenience.
~~~

There are 50 random tests with upper bound of the input = 1000.


## Examples

When the input is 100, you need to return `3`, since that is the minimum number of signs required, because: 123 - 45 - 67 + 89 = 100 (3 operators in total).

More examples:
```
 11  -->  5  #  1 + 2 + 34 + 56 + 7 - 89 = 11
100  -->  3  #  123 - 45 - 67 + 89 = 100
766  -->  4  #  1 - 2 + 34 - 56 + 789 = 766
160  -->  -  #  no solution possible
```

Inspired by a [puzzle on BBC Radio 4](https://www.bbc.co.uk/programmes/p057wxwl) (which is unfortunately not available anymore)

```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 operator_insertor(total):\n  q = [(total, "123456789", 0)]\n  while q:\n    s, n, d = q.pop(0)\n    if int(n) == s: return d\n    for i in range(0, len(n) - 1):\n        n1 = int(n[0:i + 1])\n        n2 = n[i + 1:]\n        q.append((s - n1, n2, d + 1))\n        q.append((n1 - s, n2, d + 1))'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'operator_insertor'
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: 201943