Two Angles, Two Lengths

Vectors $\vec{a}, \vec{b}$ make an angle of $\dfrac{\pi}{3}$ and $\left|\vec{a} - \vec{b}\right| = 5$; the vectors $\vec{c} - \vec{a}$ and $\vec{c} - \vec{b}$ make an angle of $\dfrac{2\pi}{3}$, and $\left|\vec{c} - \vec{a}\right| = 2\sqrt{3}$. Find the maximum value of $\vec{a}\cdot\vec{c}$.