
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 two fixed points $A(0,-2)$ and $B(0,2)$, point $P$ is on the ellipse $\frac{x^2 }{12}+ \frac{y^2 }{16}=1$ and satisfies $|{\overrightarrow{AP}}|-|{\overrightarrow{BP}}|=2$. Find $\overrightarrow{AP} \cdot \overrightarrow{BP}$.

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