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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