Trigonometry · Superchallenge problem

Vectors $\mathbf a$ and $\mathbf b$ in the plane satisfy $|\mathbf a| = 2$, $|\mathbf b| = 3$, and $\mathbf a\cdot\mathbf b = 3$. A vector $\mathbf c$ satisfies $(\mathbf c - \mathbf a)\cdot(\mathbf c - \mathbf b) = 0$. Find the largest possible value of $|\mathbf c|$.