
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 \( C \), and \( P \) is a point on the hyperbola in the first quadrant. The centroid of \(\triangle P F_1 F_2\) is \( G \) and the incenter is \( I \).
(1) Does there exist a point \( P \) such that \( IG \parallel F_1 F_2 \)?
(2) Let \( A \) be the left vertex of the hyperbola \( C \), and \( l \) be a line passing through the right focus \( F_2 \), intersecting the hyperbola \( C \) 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 the line \( l \).

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