CF1198D Rectangle Painting 1

There is a square grid of size $ n \times n $ . Some cells are colored in black, all others are colored in white. In one operation you can select some rectangle and color all its cells in white. It costs $ \max(h, w) $ to color a rectangle of size $ h \times w $ . You are to make all cells white for minimum total cost.

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The examples and some of optimal solutions are shown on the pictures below.