Cosines in a Triangle

In $\triangle ABC$, $\cos A=\frac13$ and $\cos B=\frac14$. Then $\cos C=\dfrac{a\sqrt{b}-c}{d}$, where $a,b,c,d$ are positive integers, $\gcd(a,c,d)=1$, and $b$ is squarefree. Find $a+b+c+d$.