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Geometry Level 4

If \({ A }_{ 1 },{ A }_{ 2 },...,{ A }_{ n }\) be the vertices of a \(n\)- sided regular polygon such that:\[\frac { 1 }{ { A }_{ 1 }{ A }_{ 2 } } =\frac { 1 }{ { A }_{ 1 }{ A }_{ 3 } } +\frac { 1 }{ { A }_{ 1 }{ A }_{ 4 } }\]Then find the value of \(n\)

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