Given an integer array arr and an integer k, modify the array by repeating it k times.
For example, if arr = [1, 2] and k = 3 then the modified array will be [1, 2, 1, 2, 1, 2].
Return the maximum sub-array sum in the modified array. Note that the length of the sub-array can be 0 and its sum in that case is 0.
As the answer can be very large, return the answer modulo 10^9 + 7.
 
Example 1:
Input: arr = [1,2], k = 3
Output: 9

Example 2:
Input: arr = [1,-2,1], k = 5
Output: 2

Example 3:
Input: arr = [-1,-2], k = 7
Output: 0

 
Constraints:

1 <= arr.length <= 10^5
1 <= k <= 10^5
-10^4 <= arr[i] <= 10^4

```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 kConcatenationMaxSum(self, arr: List[int], k: int) -> int:\n        # 1 -2 1  1 -2 1   1 -2 1   1 -2 1   1 -2 1\n        if max(arr) <= 0:\n            return 0\n        if sum(arr) > 0:        # 需要找到最大的起始点\n            res = sum(arr) * (k - 2)\n            dp = [0] * len(arr)\n            dp[0] = arr[0]\n            for i in range(1,len(arr)):\n                dp[i] = dp[i - 1] + arr[i]\n                    \n            res += max(dp)\n            dp = [0] * len(arr)\n            dp[-1] = arr[-1]\n            for i in range(len(arr) - 2, -1, -1):\n                dp[i] = dp[i + 1] + arr[i]\n                    \n            res += max(dp)\n            return (res)% (10 ** 9 + 7)\n        \n        # 如果sum <= 0 那么只需要考虑两轮就好了\n        cur , res = 0 , 0\n        arr = arr + arr\n        for num in arr:\n            cur = max(cur+num,num)\n            res = max(res,cur)\n        \n        return res % (10 ** 9 + 7)\n        \n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'kConcatenationMaxSum'
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: 6036