Two perpendicular chords of a circle \(A_{1}A_{4}\) and \(A_{2}A_{3}\) intersect at a point \(P\). If for \(i=1, 2, 3\), the length of \(PA_{i}=2^{i}\), the radius of the circle can be expressed as \(\sqrt{a}\) where \(a\) is a positive integer, find \(a\).

This problem is part of the set Intermediate Problems

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