An unrelated triangle

Geometry Level 4

Points \(A, B,\) and \(C\) are lying on a circle centered at \(P\) such that \(AC\) intersects \(BP\) at \(D,\) where \(AD=7, \angle APB\, (\alpha) = 120^{\circ}, \) and \(3[ADB]= 2[ADP].\)

If the area of \(\triangle CPD \) can be expressed in the form \( \dfrac{a \sqrt{b}}{c} \) for positive integers \(a, b,\) and \(c\) such that \(a,c\) are coprime, and \(b \) is square-free, determine \( a + b + c. \)

Notation: \([\,\cdot\,]\) denotes the area of the figure.


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