Slid sectors

Geometry Level 3

A circle of radius \(20\) is sliced into three congruent sectors, which are then slid apart to create a green equilateral triangle with side length \(9.\) A larger equilateral triangle is then circumscribed.

The side length of this large triangle can be written as \(\frac{a+b\sqrt{d}}{c}\) with \(d\) square-free and \(a, b, c\) irreducible.

What is the value of \(a+b+c+d?\)

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