
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 \( P \) be an arbitrary point on the graph of the function \( y = x + \frac{2}{x} \) (for \( x > 0 \)). From point \( P \), perpendiculars are drawn to the line \( y = x \) and to the y-axis, with the feet of these perpendiculars labeled as \( A \) and \( B \), respectively. Find the value of \( \overrightarrow{P A} \cdot \overrightarrow{P B} \).

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