The Tangent Triangle Identity

In $\triangle ABC$, $\tan A=6$ and $\tan B=7$. Find $\tan A\cdot\tan B\cdot\tan C$. The answer is in the form $\frac{\alpha}{\beta}$ where $\alpha$ and $\beta$ are positive integers, and $\gcd(\alpha,\beta) = 1$. Input the value of $\alpha + \beta$.