There are two types of soup: type A and type B. Initially we have N ml of each type of soup. There are four kinds of operations:

Serve 100 ml of soup A and 0 ml of soup B
Serve 75 ml of soup A and 25 ml of soup B
Serve 50 ml of soup A and 50 ml of soup B
Serve 25 ml of soup A and 75 ml of soup B

When we serve some soup, we give it to someone and we no longer have it.  Each turn, we will choose from the four operations with equal probability 0.25. If the remaining volume of soup is not enough to complete the operation, we will serve as much as we can.  We stop once we no longer have some quantity of both types of soup.
Note that we do not have the operation where all 100 ml's of soup B are used first.  
Return the probability that soup A will be empty first, plus half the probability that A and B become empty at the same time.
 
Example:
Input: N = 50
Output: 0.625
Explanation: 
If we choose the first two operations, A will become empty first. For the third operation, A and B will become empty at the same time. For the fourth operation, B will become empty first. So the total probability of A becoming empty first plus half the probability that A and B become empty at the same time, is 0.25 * (1 + 1 + 0.5 + 0) = 0.625.


Notes: 

0 <= N <= 10^9. 
Answers within 10^-6 of the true value will be accepted as correct.

```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 soupServings(self, N: int) -> float:\n        import numpy as np\n        if N > 5000: return 1\n        if N == 0:\n            return 0.5\n        if N % 25 == 0:\n            N = N // 25\n        else:\n            N = N // 25 + 1\n        \n        # N is the dimension\n        # Initial Square equals to ( N + 1 ) * ( N + 1 )\n        Matrix = [ [ 0.0 for i in range( N + 1 ) ] for j in range( N + 1 ) ]\n        \n        Matrix = np.matrix( Matrix )\n        Matrix[ N, N ] = 1\n        \n        Explore_List = [ (N,N) ];\n        while( Explore_List ):\n            New_List = set()\n            for x, y in Explore_List:\n                if( x != 0 and y != 0 ):\n                    Matrix[ max( 0, x - 2 ), max( 0, y - 2 ) ] += Matrix[ x, y ] * 1/4\n                    New_List.add( (max( 0, x - 2 ), max( 0, y - 2 )) )\n                    Matrix[ max( 0, x - 1 ), max( 0, y - 3 ) ] += Matrix[ x, y ] * 1/4\n                    New_List.add( (max( 0, x - 1 ), max( 0, y - 3 )) )\n                    Matrix[ max( 0, x - 3 ), max( 0, y - 1 ) ] += Matrix[ x, y ] * 1/4\n                    New_List.add( (max( 0, x - 3 ), max( 0, y - 1 )) )\n                    Matrix[ max( 0, x - 4 ), max( 0, y - 0 ) ] += Matrix[ x, y ] * 1/4\n                    New_List.add( (max( 0, x - 4 ), max( 0, y - 0 )) )\n\n            Explore_List = New_List\n            \n        return Matrix[ 0 ].sum() - 1/2 * Matrix[ 0, 0 ]\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'soupServings'
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: 6171316