
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 $A$, $B$, and $P$ are three distinct points on the hyperbola ${x^2}-\frac{{y^2}}{4}=1$, and they satisfy $\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PO}$ (where $O$ is the origin), the slopes of lines $PA$ and $PB$ are denoted as $m$ and $n$ respectively. Find the minimum value of ${m^2}+\frac{{n^2}}{9}$.

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