Trigonometry · Superchallenge problem

Vectors $\mathbf a$ and $\mathbf b$ have $|\mathbf a| = 3$ and $|\mathbf b| = 4$, and the angle between them is at most $60^\circ$. Find the largest possible value of $|\mathbf a + \mathbf b| + |\mathbf a - \mathbf b|$, in simplest radical form.