Nice Angles, But Not-So-Nice Answer

Geometry Level 5

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

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