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Geometry Level 4

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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