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Geometry Level 3

Suppose A1A_{1}A2A_{2}A3A_{3}\ldotsAnA_{n} is an nn-sided regular polygon such that

1A1A2=1A1A3+1A1A4.\frac{1}{A_{1}A_{2}}=\frac{1}{A_{1}A_{3}}+\frac{1}{A_{1}A_{4}}.

Determine the number of sides of the polygon.

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