Trigonometry · Superchallenge problem

Vectors $\mathbf a$ and $\mathbf b$ have $|\mathbf a| = 1$ and $|\mathbf b| = 2$, and the angle between them is $60^\circ$. As $t$ ranges over all real numbers, find the minimum value of $|t\mathbf a - \mathbf b| + \left|t\mathbf a - \tfrac12\mathbf b\right|$.