From Start to Start to Start to Start...

Before Alice and Bob head to the park, they consider the following series of separate regular polygons, where their inner polygons intersect at the midpoints. For each regular polygon, the Start point is indicated by the white dot.

As usual, Alice and Bob both start at the white dot. She prefers to run around the outer polygon, and he prefers to run around the inner polygon, both at the same speed. They want to get back to the Start point together after a few laps.

Which type of park shapes should they choose?

Note: Circle is not included in the polygon list.

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This is the extension of the previous problem .