In the coordinate plane, for a parallelogram OACB with $\overline { \mathrm { OA } } = \sqrt { 2 } , \overline { \mathrm { OB } } = 2 \sqrt { 2 }$ and $\cos ( \angle \mathrm { AOB } ) = \frac { 1 } { 4 }$, point P satisfies the following conditions. (가) $\overrightarrow { \mathrm { OP } } = s \overrightarrow { \mathrm { OA } } + t \overrightarrow { \mathrm { OB } } ( 0 \leq s \leq 1, 0 \leq t \leq 1 )$ (나) $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OB } } + \overrightarrow { \mathrm { BP } } \cdot \overrightarrow { \mathrm { BC } } = 2$ For a point X moving on a circle centered at O and passing through point A, let $M$ and $m$ be the maximum and minimum values of $| 3 \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OX } } |$ respectively. When $M \times m = a \sqrt { 6 } + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Given that $a$ and $b$ are rational numbers.) [4 points]
In the coordinate plane, for a parallelogram OACB with $\overline { \mathrm { OA } } = \sqrt { 2 } , \overline { \mathrm { OB } } = 2 \sqrt { 2 }$ and $\cos ( \angle \mathrm { AOB } ) = \frac { 1 } { 4 }$, point P satisfies the following conditions.\\
(가) $\overrightarrow { \mathrm { OP } } = s \overrightarrow { \mathrm { OA } } + t \overrightarrow { \mathrm { OB } } ( 0 \leq s \leq 1, 0 \leq t \leq 1 )$\\
(나) $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OB } } + \overrightarrow { \mathrm { BP } } \cdot \overrightarrow { \mathrm { BC } } = 2$
For a point X moving on a circle centered at O and passing through point A, let $M$ and $m$ be the maximum and minimum values of $| 3 \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OX } } |$ respectively. When $M \times m = a \sqrt { 6 } + b$, find the value of $a ^ { 2 } + b ^ { 2 }$.
(Given that $a$ and $b$ are rational numbers.) [4 points]