
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 a rectangular coordinate system, consider the hyperbola \(\Gamma: \frac{x^{2}}{3}-y^{2}=1\). For any point \(P\) in the plane that is not on \(\Gamma\), let \(\Omega_{P}\) be the set of lines passing through \(P\) and intersecting \(\Gamma\) at two points. For any line \(l \in \Omega_{P}\), let \(M\) and \(N\) be the two intersection points of \(l\) with \(\Gamma\), and define \(f_{P}(l) = |PM| \cdot |PN|\). If there exists a line \(l_{0} \in \Omega_{P}\) such that the two intersections of \(l_{0}\) with \(\Gamma\) lie on opposite sides of the y-axis, and for any other line \(l \in \Omega_{P}\), \(l \neq l_{0}\), we have \(f_{P}(l) > f_{P}(l_{0})\), then \(P\) is called a "good point". Find the area of the region formed by all good points.

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