
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 the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with the left vertex at $A_1$ and the right focus at $F_2$, and let $P$ be a point on the right branch of the hyperbola, find the minimum value of $\overrightarrow{PA_1} \cdot \overrightarrow{PF_2}$.

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