Partition and Maximize

You are given an array $A$ of size $N$. Consider the following steps: - Choose an integer $k$ $(1 \leq k \leq N)$; - Divide the range $[1, N]$ into $k$ intervals $[l_i,r_i]$ $(1 \leq i \le k)$ such that: - $l_1=1$ and $r_k=N$; - $l_{i+1}=r_i+1$ for all $(1 \leq i \lt k)$. - The score of this split is defined as $k-\sum_{i=1}^k \max(A_{l_i},A_{l_{i}+1},\ldots,A_{r_i})$. Find the **maximum