In the coordinate plane, there is a square ABCD with side length 4. $$|\overrightarrow{\mathrm{XB}} + \overrightarrow{\mathrm{XC}}| = |\overrightarrow{\mathrm{XB}} - \overrightarrow{\mathrm{XC}}|$$ Let $S$ be the figure formed by points X satisfying this condition. For a point P on figure $S$, $$4\overrightarrow{\mathrm{PQ}} = \overrightarrow{\mathrm{PB}} + 2\overrightarrow{\mathrm{PD}}$$ Let Q be the point satisfying this condition. If the maximum and minimum values of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{AQ}}$ are $M$ and $m$ respectively, find the value of $M \times m$. [4 points]
In the coordinate plane, there is a square ABCD with side length 4.
$$|\overrightarrow{\mathrm{XB}} + \overrightarrow{\mathrm{XC}}| = |\overrightarrow{\mathrm{XB}} - \overrightarrow{\mathrm{XC}}|$$
Let $S$ be the figure formed by points X satisfying this condition. For a point P on figure $S$,
$$4\overrightarrow{\mathrm{PQ}} = \overrightarrow{\mathrm{PB}} + 2\overrightarrow{\mathrm{PD}}$$
Let Q be the point satisfying this condition. If the maximum and minimum values of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{AQ}}$ are $M$ and $m$ respectively, find the value of $M \times m$. [4 points]