In coordinate space, there are two points $\mathrm { P } , \mathrm { Q }$ moving on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$. Let $\mathrm { P } _ { 1 } , \mathrm { Q } _ { 1 }$ be the feet of the perpendiculars from points P and Q to the plane $y = 4$ respectively, and let $\mathrm { P } _ { 2 } , \mathrm { Q } _ { 2 }$ be the feet of the perpendiculars to the plane $y + \sqrt { 3 } z + 8 = 0$ respectively. Find the maximum value of $2 | \overrightarrow { \mathrm { PQ } } | ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 1 } \mathrm { Q } _ { 1 } } \right| ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 2 } \mathrm { Q } _ { 2 } } \right| ^ { 2 }$. [4 points]