Maximizing a Variable in a Quadratic Constraint

If $a$ and $b$ are real numbers such that $3a^2 + b^2 = 12a-4b$, what is the greatest possible value of $a$? The answer is in the form $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, $c$ are positive integers. Input the smallest value of $a+b+c$.