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 findRepSubstrGivenLength(self, nums: List[int], l:int, base:int, modulus:int) -> int:\n        \n        n = len(nums)\n        \n        h = 0\n        for i in range(l):\n            h = (h * base + nums[i]) % modulus\n            \n        seen = {h}\n            \n        for start in range(1, n-l+1):\n            h = (h * base - nums[start - 1] * pow(base, l, modulus) + nums[start + l - 1]) % modulus\n            \n            if h in seen:\n                return start\n            \n            seen.add(h)\n            \n        return -1    \n    \n    \n    def longestDupSubstring(self, S: str) -> str:\n        if not S:\n            return \'\'\n                \n        nums = [ord(S[i]) - ord(\'a\') for i in range(len(S))]\n        \n        base = 26\n        modulus = 2**32\n        \n        left = 1\n        right = len(S)\n        \n        while left <= right:\n            mid = left + (right - left) // 2\n            \n            if self.findRepSubstrGivenLength(nums, mid, base, modulus) != -1:\n                left = mid + 1\n            else:\n                right = mid -1\n                \n        start = self.findRepSubstrGivenLength(nums, left-1, base, modulus)\n        \n        return S[start: start + left -1]    \n        \n        \n        \n\'\'\'\nclass Solution:\n    class Node:\n        def __init__(self, v=\'*\'): \n            self.value = v\n            self.next = dict()\n            self.occurences = set()\n                   \n                \n    def print_tree(self, node, indentation):\n        print(indentation + str(node.value) + \\": \\" + str(node.occurences))\n\n        for v in node.next.keys():\n            self.print_tree(node.next[v], indentation + \'\\\\t\')\n        \n        \n    def getNode(self, node, v):\n        if v not in node.next.keys():\n            node.next[v] = self.Node(v)\n\n        return node.next[v]\n\n\n    def AddOccurences(self, node, i, S):\n        node.occurences.add(i)\n\n        if len(node.occurences) == 2:\n            for o in node.occurences:\n                if o + 1 < len(S):\n                    self.AddOccurences(self.getNode(node, S[o+1]), o+1, S)\n\n\n        if len(node.occurences)>2:\n            if i + 1 < len(S):\n                self.AddOccurences(self.getNode(node, S[i+1]), i+1, S)\n\n        \n    def getLongestDuplicate(self, node):\n        maxSubstring = \'\'\n        \n        for v in node.next.keys():\n            if len(node.next[v].occurences)>1:\n\n                childSubstring = self.getLongestDuplicate(node.next[v])\n                if len(childSubstring) + 1 > len(maxSubstring):\n                    maxSubstring = v + childSubstring\n                        \n        return maxSubstring\n    \n    \n    def longestDupSubstring(self, S: str) -> str:\n        if not S:\n            return 0\n        \n        if len(set(S)) == 1:\n            return S[0:len(S)-1]\n\n        root = self.Node()\n\n        for i in range(len(S)):\n            self.AddOccurences(self.getNode(root, S[i]), i, S)\n\n        #self.print_tree(root, \'\')\n        maxDepthDuplicates = self.getLongestDuplicate(root)\n        \n        return maxDepthDuplicates\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: 6960