Maximal Prefix After Flip

You are given an array $A$ consisting of $N$ integers. For each $K$ from $0$ to $N$ find the **maximum** prefix sum of $A$ after changing the sign of **at most** $K$ elements of the array. Formally: Let $F(A)$ denote the **maximum** prefix sum of an array $A$ of length $N$ (including an empty array for which prefix sum is $0$). Thus, $F(A)$ = $\max(0,\ \max_{1 \le i \le N} \sum_{j=1}^{i}A_j )$