# Ratio of Ratios

Geometry Level 4

$\dfrac{2 \sin A + 3 \sin B}{38}=\dfrac{4 \sin B + 5\sin C}{77}=\dfrac{5\sin A + 2\sin C}{53}$

Let $$A, B, C$$ be angles of a triangle that satisfy the condition above.

The ratio $$\cos A : \cos B : \cos C$$ is in the reduced form $$x:y:z$$ where $$x,y,z$$ are positive integers with $$\gcd(x,y,z)=1$$. What is the value of $$x+y+z$$?

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