Two perpendicular chords of a circle \(A_{1}A_{4}\) and \(A_{2}A_{3}\) intersect at a point \(P\). If for \(i=1, 2, 3, 4\), the length of \(PA_{i}=i\), the radius of the circle can be expressed as \( \frac{\sqrt{a}}{b} \) where \(a, b\) are positive integers with \(a\) square-free. Find \(ab\).

This problem is part of the set Intermediate Problems

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