Given a string S, consider all duplicated substrings: (contiguous) substrings of S that occur 2 or more times.  (The occurrences may overlap.)
Return any duplicated substring that has the longest possible length.  (If S does not have a duplicated substring, the answer is "".)
 
Example 1:
Input: "banana"
Output: "ana"

Example 2:
Input: "abcd"
Output: ""

 
Note:

2 <= S.length <= 10^5
S consists of lowercase English letters.

```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 longestDupSubstring(self, S: str) -> str:\n        def checkSubstrings(size):\n            hashed = 0\n            pow_mult = pow(26, size) % mod\n            \n            for i in range(size):\n                hashed *= 26\n                hashed += char_vals[i]\n            \n            hashed %= mod\n            seen = {hashed}\n            \n            for i in range(size, len(S)):\n                hashed = (hashed * 26 - char_vals[i-size] * pow_mult + char_vals[i]) % mod\n                if hashed in seen:\n                    return i-size+1\n                seen.add(hashed)\n            \n            return None\n\n        s, f = 0, len(S)\n        longest = 0\n        \n        char_vals = [ord(c) - ord('a') for c in S]\n        mod = pow(2, 63) - 1\n        \n        while s < f:\n            m = s + (f-s+1) // 2\n            longest_check = checkSubstrings(m)\n            if longest_check:\n                longest = longest_check\n                s = m\n            else:\n                f = m-1\n        \n        return S[longest:longest+s]\n"
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'longestDupSubstring'
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: 7182