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

$$\begin{align} & x_k = cos(\frac{2k - 1}{n} × 180°) \\ & y_k = sin(\frac{2k - 1}{n} × 180°) \end{align}$$

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 $S_3$ and $S_4$ is the six-sided shape shown in pink below:

How many sides does $S_{1864} + S_{1865} + \ldots + S_{1909}$ have?

--hints--

minkowskiSums() should return 86226.

assert.strictEqual(minkowskiSums(), 86226);

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function minkowskiSums() {

  return true;
}

minkowskiSums();

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// solution required