Let the mean and standard deviation of marks of class $A$ of $100$ students be respectively $40$ and $\alpha\ (>\,0)$, and the mean and standard deviation of marks of class $B$ of $n$ students be respectively $55$ and $30-\alpha$. If the mean and variance of the marks of the combined class of $100+n$ students are respectively $50$ and $350$, then the sum of variances of classes $A$ and $B$ is: