In the coordinate plane, there is an equilateral triangle ABC with side length 4. Let D be the point that divides segment AB in the ratio $1:3$, E be the point that divides segment BC in the ratio $1:3$, and F be the point that divides segment CA in the ratio $1:3$. Four points $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$, and $\mathrm{X}$ satisfy the following conditions. (가) $|\overrightarrow{\mathrm{DP}}| = |\overrightarrow{\mathrm{EQ}}| = |\overrightarrow{\mathrm{FR}}| = 1$ (나) $\overrightarrow{\mathrm{AX}} = \overrightarrow{\mathrm{PB}} + \overrightarrow{\mathrm{QC}} + \overrightarrow{\mathrm{RA}}$ When $|\overrightarrow{\mathrm{AX}}|$ is maximized, let the area of triangle PQR be $S$. Find the value of $16S^2$. [4 points]
In the coordinate plane, there is an equilateral triangle ABC with side length 4. Let D be the point that divides segment AB in the ratio $1:3$, E be the point that divides segment BC in the ratio $1:3$, and F be the point that divides segment CA in the ratio $1:3$. Four points $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$, and $\mathrm{X}$ satisfy the following conditions.\\
(가) $|\overrightarrow{\mathrm{DP}}| = |\overrightarrow{\mathrm{EQ}}| = |\overrightarrow{\mathrm{FR}}| = 1$\\
(나) $\overrightarrow{\mathrm{AX}} = \overrightarrow{\mathrm{PB}} + \overrightarrow{\mathrm{QC}} + \overrightarrow{\mathrm{RA}}$\\
When $|\overrightarrow{\mathrm{AX}}|$ is maximized, let the area of triangle PQR be $S$. Find the value of $16S^2$. [4 points]