The aim is to calculate `exponential(x)` (written `exp(x)` in most math libraries) as an irreducible fraction, the numerator of this fraction having a given number of digits.

We call this function `expand`, it takes two parameters, `x` of which we want to evaluate the exponential, `digits` which is the required number of digits for the numerator. 

The function `expand` will return an array of the form `[numerator, denominator]`; we stop the loop in the Taylor expansion (see references below) when the numerator has a number of digits `>=` the required number of digits

# Examples:
```
expand(1, 2) --> 65/24 (we will write this [65, 24] or (65, 24) in Haskell; 
65/24 ~ 2.708...)

expand(2, 5) --> [20947, 2835]

expand(3, 10) --> [7205850259, 358758400]

expand(1.5, 10) --> [36185315027,8074035200]
```
**Note**
```expand(1,5) = [109601, 40320]``` is the same as ```expand(1, 6)```

#Method:
As said above the way here is to use `Taylor expansion` of the exponential function though it is not always the best approximation by a rational.

#References:
https://en.wikipedia.org/wiki/Exponential_function#Formal_definition

http://www.efunda.com/math/taylor_series/exponential.cfm

```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 fractions import Fraction, gcd\nfrom math import floor\nfrom decimal import Decimal\ndef expand(x, digit):\n    x = Fraction(Decimal(str(x)))\n    res = Fraction(1)\n    new = Fraction(1)\n    exponent = 0\n    while len(str(res.numerator)) < digit:\n        exponent += 1\n        new *= x / exponent\n        res += new\n    return [res.numerator, res.denominator]'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'expand'
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: