Extreme rationalizing 2

Algebra Level 4

A=3+52+3+5+35235A=\dfrac{3+\sqrt{5}}{\sqrt{2}+\sqrt{3+\sqrt{5}}}+\dfrac{3-\sqrt{5}}{\sqrt{2}-\sqrt{3-\sqrt{5}}}

If AA can be expressed as a+bca+b\sqrt{c} for non-negative integers a,b,ca,b,c with cc square free, then find the value of a+b+ca+b+c.

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