# Task

Christmas is coming. In the [previous kata](https://www.codewars.com/kata/5a405ba4e1ce0e1d7800012e), we build a custom Christmas tree with the specified characters and the specified height.

Now, we are interested in the center of the Christmas tree. 

Please imagine that we build a Christmas tree with `chars = "abc" and n = Infinity`, we got:
```
          a
         b c
        a b c
       a b c a
      b c a b c
     a b c a b c
    a b c a b c a
   b c a b c a b c
  a b c a b c a b a
 b c a b c a b a b c
a b c a b a b c a b c
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
          |
          |
          .
          .
```
If we keep only the center part of leaves, we will got:
```
          a
            
          b  
              
          a    
                
          a        
                    
          b        
                    
          a          
          .
          
          .
          
          .
```
As you can see, it's a infinite string, but it has a repeat substring "aba"(spaces will be removed). If we join them together, it looks like:`"abaabaabaabaaba......"`.

So, your task is to find the repeat substring of the center part of leaves.

# Inputs:

- `chars`: the specified characters. In this kata, they always be lowercase letters.

# Output:

- The repeat substring that satisfy the above conditions.

Still not understand the task? Look at the following example ;-)

# Examples


For `chars = "abc"`,the output should be `"aba"`
```
center leaves sequence: "abaabaabaabaabaaba....."
```
For `chars = "abab"`,the output should be `a`
```
center leaves sequence: "aaaaaaaaaa....."
```
For `chars = "abcde"`,the output should be `aecea`
```
center leaves sequence: "aeceaaeceaaecea....."
```
For `chars = "aaaaaaaaaaaaaa"`,the output should be `a`
```
center leaves sequence: "aaaaaaaaaaaaaa....."
```
For `chars = "abaabaaab"`,the output should be `aba`
```
center leaves sequence: "abaabaabaaba....."
```

```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 = '# I like to think of this in terms of math but it\'s been a while\n# since number theory and modular arithmatic. Looking at this, we\n# get the serries 1, 5, 13, 25, 41,... as we\'re starting at 0,\n# it\'s actually 0, 4, 12, 24, 40, ... So this is 4*sum of the first\n# n-1 numbers which gives us the formula of the nth number (starting\n# with 0) is 2n(n+1).\n\n# So the the nth character in the center is 2n(n+1) % len(chars).\n# But this is an infinite serries so I need to know when to stop.\n# If n is odd, then the pattern repeats every n (only half + 1 letters)\n# If n is even, then the pattern repeats ever n/2 generally (sometimes shorter).\n\n# Need to check the final product to make sure that there is no repeat\n# in there as the exceptions in the evens and the characters may not be unique.\ndef center_of(chars):\n    center = ""\n    n = len(chars)\n    if n == 0:\n        return center\n    elif n % 2 == 0:\n        n //= 2\n    # get the list of character positions\n    pos = []\n    for i in range(n):\n        pos.append(2*i*(i+1) % len(chars))\n    # get the characters in the center\n    for num in pos:\n        center += chars[num]\n    # test if there\'s are repeat in the center\n    for i in range(len(center)//2):\n        test = center[:i+1]\n        if center == test * (len(center)//len(test)):\n            center = test\n            break  \n    return center'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'center_of'
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: 5686