# Nice Angles, But Not-So-Nice Answer

Geometry Level 4

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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