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 square-free integer. Find $$a$$.