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        \n        def code(t):\n            return ord(t) - ord('a')+1\n        def check(length):\n            seen = collections.defaultdict(list)\n            \n            MOD = 10 ** 9 + 7\n            P = 113\n            INV_P = pow(P, MOD-2, MOD)\n\n            h = 0\n            power = 1\n            '''\n            h1 = s[0]p^0+s[1]p^1+s[2]p^2 +...+s[length-1]p^(length-1)\n            move next:\n            h2 = s[1]p^0+s[2]p^1+s[3]p^2 +...+s[length]p^(length-1)\n            \n            h2 = (h1 - S[i-length+1])*INV_P % MOD\n            \n            '''\n            for i, x in enumerate(S):\n                h = (h + power*code(x)) % MOD \n                if i < length-1:\n                    power = power * P % MOD\n                else:\n                    # start: i-(length-1)\n                    if h in seen:\n                        for j in seen[h]:\n                            if S[i-(length-1):i+1] == S[j:j+length]:\n                                return S[j:j+length], True\n                    seen[h].append(i-(length-1))\n                    h = (h-code(S[i-(length-1)]))*INV_P % MOD\n            \n            return '', False\n        \n        res = ''\n        l, r = 1, len(S)-1\n        while l <= r:\n            mid = (l+r)//2\n            sub, is_check = check(mid)\n            if is_check:\n                res = sub\n                l = mid + 1\n            else:\n                r = mid - 1\n        \n        return res"
_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: 7879