Nice Angles, But Not-So-Nice Answer

Geometry Level 5

Four points \(A,B,C,D\) satisfy \(AB=1\), \(BC=2\), \(CD=3\), \(\angle ABC=45^{\circ}\), and \(\angle BCD=60^{\circ}\). The largest possible value of \(AD\) can be expressed in the form \(\sqrt{a+\sqrt{b-\sqrt{c}}}\) for positive integers \(a,b,c\). Find \(a+b+c\).

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