For two points $\mathrm { A } ( 2 , \sqrt { 2 } , \sqrt { 3 } )$ and $\mathrm { B } ( 1 , - \sqrt { 2 } , 2 \sqrt { 3 } )$ in coordinate space, point P satisfies the following conditions. (가) $| \overrightarrow { \mathrm { AP } } | = 1$ (나) The angle between $\overrightarrow { \mathrm { AP } }$ and $\overrightarrow { \mathrm { AB } }$ is $\frac { \pi } { 6 }$. For point Q on a sphere centered at the origin with radius 1, the maximum value of $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } }$ is $a + b \sqrt { 33 }$. Find the value of $16 \left( a ^ { 2 } + b ^ { 2 } \right)$. (Here, $a$ and $b$ are rational numbers.) [4 points]
For two points $\mathrm { A } ( 2 , \sqrt { 2 } , \sqrt { 3 } )$ and $\mathrm { B } ( 1 , - \sqrt { 2 } , 2 \sqrt { 3 } )$ in coordinate space, point P satisfies the following conditions.\\
(가) $| \overrightarrow { \mathrm { AP } } | = 1$\\
(나) The angle between $\overrightarrow { \mathrm { AP } }$ and $\overrightarrow { \mathrm { AB } }$ is $\frac { \pi } { 6 }$.
For point Q on a sphere centered at the origin with radius 1, the maximum value of $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } }$ is $a + b \sqrt { 33 }$. Find the value of $16 \left( a ^ { 2 } + b ^ { 2 } \right)$. (Here, $a$ and $b$ are rational numbers.) [4 points]