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 RabinKarp(text, M, q):\n        #     if M == 0: return True\n        #     h, t, d = (1<<(8*M-8))%q, 0, 256\n        #     dic = defaultdict(list)\n        #     for i in range(M): \n        #         t = (d * t + ord(text[i]))% q\n        #     dic[t].append(i-M+1)\n        #     for i in range(len(text) - M):\n        #         t = (d*(t-ord(text[i])*h) + ord(text[i + M]))% q\n        #         for j in dic[t]:\n        #             if text[i+1:i+M+1] == text[j:j+M]:\n        #                 return (True, text[j:j+M])\n        #         dic[t].append(i+1)\n        #     return (False, \\"\\")\n\n\n        def check(sz):\n            seen = defaultdict(list)\n            cur, base, MOD = 0, 256, (1<<31) - 1\n            h = (1 << (sz*8)) % MOD\n            for i in range(sz):\n                cur *= base\n                cur += ord(S[i])\n                cur %= MOD\n            seen[cur].append(0)\n            for i in range(sz,len(S)):\n                cur *= base\n                cur += ord(S[i])\n                cur -= ord(S[i-sz])*h \n                cur %= MOD\n                for j in seen[cur]:\n                    if S[j:j+sz] == S[i-sz+1:i+1]:\n                        return True, S[i-sz+1:i+1]\n                seen[cur].append(i-sz+1)\n            return False, \'\'\n\n        lo, hi = 1, len(S)\n        res = \'\'\n        # MOD = (1<<31) - 1\n        while lo <= hi:\n            mid = (lo+hi)//2\n            flag, tmp = check(mid)\n            # flag, tmp = RabinKarp(S, mid, q)\n            if flag:\n                lo = mid + 1\n                res = tmp\n                # print(mid,res)\n            else:\n                hi = mid - 1\n        # print(lo,hi)\n        return res \n\n\n# class Solution:\n#     def RabinKarp(self,text, M, q):\n#         if M == 0: return True\n#         h, t, d = (1<<(8*M-8))%q, 0, 256\n\n#         dic = defaultdict(list)\n\n#         for i in range(M): \n#             t = (d * t + ord(text[i]))% q\n\n#         dic[t].append(i-M+1)\n\n#         for i in range(len(text) - M):\n#             t = (d*(t-ord(text[i])*h) + ord(text[i + M]))% q\n#             for j in dic[t]:\n#                 if text[i+1:i+M+1] == text[j:j+M]:\n#                     return (True, text[j:j+M])\n#             dic[t].append(i+1)\n#         return (False, \\"\\")\n\n#     def longestDupSubstring(self, S):\n#         beg, end = 0, len(S)\n#         q = (1<<31) - 1 \n#         Found = \\"\\"\n#         while beg + 1 < end:\n#             mid = (beg + end)//2\n#             isFound, candidate = self.RabinKarp(S, mid, q)\n#             if isFound:\n#                 beg, Found = mid, candidate\n#             else:\n#                 end = mid\n\n#         return Found\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: 7483