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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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