Can you solve without trig?

Geometry Level 4

Triangle \(ABC\) inscribed a circle \((O;1)\). If \(\angle A = 60 ^\circ\) and \(\angle C = 45 ^\circ\), 2 times the perimeter of the triangle can be written as \(a \sqrt{m} + b\sqrt{n} + c\sqrt{p}\), where \(a,b,c,m,n,p\) are positive integers and \(m,n,p\) are distinct squarefrees. Find \(abc+mnp\).

Note: 'Cause I don't like trigonometry, please post solutions using only trigonometric functions of these angles: \(30 ^\circ,45 ^\circ,60 ^\circ\)

Inspired by my math exercise.

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