Let S_n be the regular n-sided polygon – or shape – whose vertices v_k (k = 1,2,…,n) have coordinates:

Each S_n is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.

The Minkowski sum, S+T, of two shapes S and T is the result of adding every point in S to every point in T, where point addition is performed coordinate-wise: (u, v) + (x, y) = (u+x, v+y).

For example, the sum of S₃ and S₄ is the six-sided shape shown in pink below:

How many sides does S₁₈₆₄ + S₁₈₆₅ + … + S₁₉₀₉ have?