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Geometry Level 3

Suppose $$A_{1}$$$$A_{2}$$$$A_{3}$$$$\ldots$$$$A_{n}$$ is an $$n$$-sided regular polygon such that

$\frac{1}{A_{1}A_{2}}=\frac{1}{A_{1}A_{3}}+\frac{1}{A_{1}A_{4}}.$

Determine the number of sides of the polygon.

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