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

$$\begin{array}{rl}{x}_k& =\cos ((2k-1)/n\times {180}^{\circ })\\ y_k& =\sin ((2k-1)/n\times {180}^{\circ })\end{array}$$

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 $S_{1864}+S_{1865}+\cdots +S_{1909}$ have?