
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that the base of the rectangular parallelepiped $ABCD-A_{1}B_{1}C_{1}D_{1}$ is a square with side length $2\sqrt{2}$, if $cos〈{\overrightarrow{AB},\overrightarrow{AC_1}}〉=\frac{{\sqrt{3}}}{3}$, then the surface area of the circumscribed sphere of this rectangular parallelepiped is ____; let $\overrightarrow{{e}_{1}}$, $\overrightarrow{{e}_{2}}$ be unit vectors in the directions of $\overrightarrow{AB}$, $\overrightarrow{AD}$, and $|\overrightarrow{a}|=2\sqrt{6}$, $\overrightarrow{a}•\overrightarrow{{e}_{1}}=\overrightarrow{a}•\overrightarrow{{e}_{2}}=2\sqrt{2}$, then the minimum value of $|\overrightarrow{a}-m\overrightarrow{{e}_{1}}-n\overrightarrow{{e}_{2}}|$ (where $m$, $n$ are constants) is ____.

Present the answer in LaTex format: \boxed{Your answer}
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