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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.