GAIC Math 2024 Problem 89

Let P be a point inside triangle ABC, and 2$\overrightarrow{PA} = \overrightarrow{PB} + \overrightarrow{PC}=0$. If $\angle BAC = \frac{\pi}{3}, BC=2$, then the maximum value of $\overrightarrow{PB}\cdot\overrightarrow{PC}$ is _____.