
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
In the Cartesian coordinate system \(xOy\), given the ellipse \(C: \frac{x^{2}}{27}+\frac{y^{2}}{18}=1\) with right focus \(F\), a line \(l\) passing through point \(F\) intersects the ellipse \(C\) at points \(A\) and \(B\). Is there a fixed point \(P\) on the \(x\)-axis such that, as the line \(l\) rotates about point \(F\), \(\overrightarrow{PA} \cdot \overrightarrow{PB}\) remains constant?

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