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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