n passengers board an airplane with exactly n seats. The first passenger has lost the ticket and picks a seat randomly. But after that, the rest of passengers will:

Take their own seat if it is still available, 
Pick other seats randomly when they find their seat occupied 

What is the probability that the n-th person can get his own seat?
 
Example 1:
Input: n = 1
Output: 1.00000
Explanation: The first person can only get the first seat.
Example 2:
Input: n = 2
Output: 0.50000
Explanation: The second person has a probability of 0.5 to get the second seat (when first person gets the first seat).

 
Constraints:

1 <= n <= 10^5

```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 nthPersonGetsNthSeat(self, n: int) -> float:\n        '''\n        f(n) = 1/n                                    -> 1st person picks his own seat\n            + 1/n * 0                                 -> 1st person picks last one's seat\n            + (n-2)/n * (                            ->1st person picks one of seat from 2nd to (n-1)th\n                1/(n-2) * f(n-1)                   -> 1st person pick 2nd's seat\n                1/(n-2) * f(n-2)                  -> 1st person pick 3rd's seat\n                ......\n                1/(n-2) * f(2)                     -> 1st person pick (n-1)th's seat\n            )\n\n        => f(n) = 1/n * ( f(n-1) + f(n-2) + f(n-3) + ... + f(1) )\n        '''\n        if n==1:\n            return 1.0\n        if n==2:\n            return 0.5\n        dp = [1.0, 0.5]\n        acc = 1.5\n        for i in range(3, n+1):\n            dp.append(acc/i)\n            acc += dp[-1]\n        return dp[-1]\n    \n        '''\n        Formal proof with math solution\n        when n > 2,\n        f(n) = 1/n * ( f(n-1) + f(n-2) + ... + f(1) )\n        f(n-1) = 1/(n-1) * (f(n-2) + f(n-3) + ... + f(1))\n\n        because, the 2nd equation requires n-1 > 1\n        So that\n\n        n * f(n) = f(n-1) + f(n-2) + f(n-3) + ... + f(1)\n        (n-1) * f(n-1) = f(n-2) + f(n-3) + ... + f(1)\n        Substract both sides, we get\n\n        n * f(n) - (n-1)*f(n-1) = f(n-1)\n\n        => n * f(n) = n * f(n-1)\n        => f(n) = f(n-1) = f(2) = 1/2 , when n > 2\n        '''\n        # return 1 if n==1 else 0.5\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'nthPersonGetsNthSeat'
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: 5436