
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
Let the ellipse be given by \(\frac{x^{2}}{5}+\frac{y^{2}}{4}=1\). The locus of the intersection points of two mutually perpendicular tangents to the ellipse is \(C\). Tangents \(PA\) and \(PB\) to the curve \(C\) intersect at point \(P\), touching the curve at points \(A\) and \(B\) respectively. Find the minimum value of \(\overrightarrow{PA} \cdot \overrightarrow{PB}\).

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