
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 hyperbola \(C: \frac{x^{2}}{4}-\frac{y^{2}}{5}=1\), \(F_{1}\) and \(F_{2}\) are the left and right foci of the hyperbola \(C\), respectively. Point \(P\) is on the hyperbola and lies in the first quadrant. The centroid and incenter of triangle \(\triangle PF_{1}F_{2}\) are \(G\) and \(I\), respectively.

1. Does there exist a point \(P\) such that \(IG \parallel F_{1}F_{2}\)?
2. Given that \(A\) is the left vertex of the hyperbola \(C\), line \(l\) passes through the right focus \(F_{2}\) and intersects the hyperbola at points \(M\) and \(N\). If the slopes \(k_{1}\) and \(k_{2}\) of \(AM\) and \(AN\) satisfy \(k_{1} + k_{2} = -\frac{1}{2}\), find the equation of line \(l\).

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