CF1697E Coloring

You are given $ n $ points on the plane, the coordinates of the $ i $ -th point are $ (x_i, y_i) $ . No two points have the same coordinates. The distance between points $ i $ and $ j $ is defined as $ d(i,j) = |x_i - x_j| + |y_i - y_j| $ . For each point, you have to choose a color, represented by an integer from $ 1 $ to $ n $ . For every ordered triple of different points $ (a,b,c) $ , the following constraints should be met: - if $ a $ , $ b $ and $ c $ have the same color, then $ d(a,b) = d(a,c) = d(b,c) $ ; - if $ a $ and $ b $ have the same color, and the color of $ c $ is different from the color of $ a $ , then $ d(a,b) < d(a,c) $ and $ d(a,b) < d(b,c) $ . Calculate the number of different ways to choose the colors that meet these constraints.

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In the first test, the following ways to choose the colors are suitable: - $ [1, 1, 1] $ ; - $ [2, 2, 2] $ ; - $ [3, 3, 3] $ ; - $ [1, 2, 3] $ ; - $ [1, 3, 2] $ ; - $ [2, 1, 3] $ ; - $ [2, 3, 1] $ ; - $ [3, 1, 2] $ ; - $ [3, 2, 1] $ .