
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. \( P \) is a point on the hyperbola in the first quadrant. The centroid of the triangle \( \triangle P F_{1} F_{2} \) is \( G \), and the incenter is \( I \).

1. Does there exist a point \( P \) such that \( I G \parallel F_{1} F_{2} \)?
2. Let \( A \) be the left vertex of the hyperbola \( C \). The line \( l \) passes through the right focus \( F_{2} \) and intersects the hyperbola \( C \) at two points \( M \) and \( N \). If the slopes of \( A M \) and \( A N \), denoted as \( k_{1} \) and \( k_{2} \), respectively, satisfy \( k_{1} + k_{2} = -\frac{1}{2} \), find the equation of the line \( l \).

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