Just be quick with calculations

Geometry Level 3

\(ABC\) is an equilateral triangle such that vertices \(B,C\) lie on two parallel lines at a distance of 6 units. If \(A\) lies between the parallel lines at a distance 4 units from one of them, then the length of the side of the triangle is of the form

\(\large{A\sqrt{\frac{B}{C}}}\), where \(A,B,C\) are co prime natural numbers.

Find \(A+B+C\).

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