Trigonometry · Superchallenge problem

Equilateral triangle $ABC$ has side length $2$, and $M$ is the midpoint of $\overline{BC}$. Point $P$ moves along the segment $\overline{AM}$. Find the minimum value of $\overrightarrow{PA}\cdot\left(\overrightarrow{PB} + \overrightarrow{PC}\right)$.