Given an integer array arr, you should partition the array into (contiguous) subarrays of length at most k. After partitioning, each subarray has their values changed to become the maximum value of that subarray.
Return the largest sum of the given array after partitioning.
 
Example 1:
Input: arr = [1,15,7,9,2,5,10], k = 3
Output: 84
Explanation: arr becomes [15,15,15,9,10,10,10]

Example 2:
Input: arr = [1,4,1,5,7,3,6,1,9,9,3], k = 4
Output: 83

Example 3:
Input: arr = [1], k = 1
Output: 1

 
Constraints:

1 <= arr.length <= 500
0 <= arr[i] <= 109
1 <= k <= arr.length

```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 = '\'\'\'\nwhat would be the state ? \ni: current position \npartition start \n\njust try allof them is probably the answer\n\n[1,15,7,9,2,5,10]\n\ntry all possible partitions and keep track of the max and sum so far \n\n\nhelper(start, i)\n\n    for j in range(i, k - i + start + 1): # these are the possible ends \n        helper(start, j)\n    \n    \nif i = 2 and start = 1 and k = 3\nfrom i=2 to 5 - 1 = 4\ninstead it should be 1, 2, 3, so only one more so i + 1 = i + k - i \n\nif i = 2 and start = 1 and k = 4\nit shld be 1, 2, 3, 4\n\ni just want to know much left to get to k \nk = end - start \nk = j + i - start -> j = k - i + start\n\nclass Solution:\n    def maxSumAfterPartitioning(self, arr: List[int], k: int) -> int:\n        def helper(i, left, curr_max, sum_so_far):\n            print(\\"helper i =\\", i, \\", left =\\", left, \\", curr_max =\\", curr_max)\n            \n            if i == len(arr):\n                self.res = max(self.res, sum_so_far + (curr_max * (k - left)))\n                return 0\n            \n            # continue current partition\n            choice1 = float(\'-inf\')\n            if left > 0:\n                choice1 = helper(i + 1, left - 1, max(curr_max, arr[i]), sum_so_far)\n            \n            # start a new partition\n            choice2 = helper(i + 1, k, arr[i], sum_so_far + (curr_max * (k - left)))\n            \n            print(\\"choice1 =\\", choice1, \\", choice2 =\\", choice2)\n            return max(choice1, choice2)\n            \n        self.res = float(\'-inf\')\n        helper(0, k, arr[0], 0)\n        return self.res\n\'\'\'\n\n\'\'\'\ndp[i]: max sum for arr ending at i \n\'\'\'\nclass Solution:\n    def maxSumAfterPartitioning(self, arr: List[int], K: int) -> int:\n        n = len(arr)\n        dp = [0 for _ in range(n+1)]\n        for i in range(n): # i is considered the end of the partition \n            currMax = 0\n            for k in range(1, min(K, i + 1) + 1):\n                currMax = max(currMax, arr[i - k + 1])\n                dp[i] = max(dp[i], dp[i-k] + (currMax * k))\n        return dp[n-1]\n\n        \n        \n        \n        \n\n\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'maxSumAfterPartitioning'
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: 5698