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 $12 - \frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $a + b?$

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