
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 point $P$ is on the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$, point $A$ satisfies $\overrightarrow{PA} = (t - 1) \overrightarrow{OP} (t \in R)$, and $\overrightarrow{OA} \cdot \overrightarrow{OP} = 64$, $\overrightarrow{OB} = (0, 1)$, find the maximum value of $|\overrightarrow{OB} \cdot \overrightarrow{OA}|$.

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