from typing import (
    List,
)

class Solution:
    """
    @param m: An integer m denotes the size of a backpack
    @param a: Given n items with size A[i]
    @return: The maximum size
    """
    def back_pack(self, m: int, a: List[int]) -> int:
        # write your code here
        # define: dp[i][j], i = this round take, j = volume
        # dp[i][j] = dp[i-1][j] # last round take
        #          = dp[i-1][j - A[i-1]] # inherit last take, we take new can succeed (only when j - A[i-1] >= 0)

        # ans: last True dp[i][volume?]

        # init: dp[all][0] = True, other False
        dp = [[False]*(m+1) for _ in range(len(a)+1)] # i = m, j = count nums
        dp[0][0] = True

        for i in range(1, len(a)+1):
            dp[i][0] = True
            for j in range(1, m + 1):
                dp[i][j] = dp[i-1][j]
                if j - a[i-1] >= 0: # can take this
                    dp[i][j] = dp[i][j] or dp[i-1][j - a[i-1]]

        # print(dp)
        for i in range(m+1):
            if dp[-1][m-i]: # check max volume
                return m-i

        return -1 # not found