Question 88, In rectangle $A B C D$, $A B=2, A D=1$, the moving point $P$ on side $D C$ (including $D, C$) and the moving point $Q$ on the extension of $C B$ (including point $\mathrm{B}$) satisfy $|\overrightarrow{\mathrm{DP}}|=|\overrightarrow{\mathrm{BQ}}|$, then the dot product $\overrightarrow{\mathrm{PA}} \cdot \overrightarrow{\mathrm{PQ}}$ of vector $\overrightarrow{\mathrm{PA}}$ and vector $\overrightarrow{\mathrm{PQ}}$ has the minimum value of $\qquad$ -