Hard to create a regular polygon

Probability Level 5

12 points are spaced evenly around the perimeter of a circle. Distinct points are chosen uniformly at random until the chosen points form the vertices of a regular polygon with at least 3 sides. In most cases, we need to pick all 12 points, in order to form a regular polygon. The expected number of points chosen is 12ab 12 - \frac{a}{b} where aa and bb are coprime positive integers. What is the value of a+b?a + b?

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