
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 the hyperbola $\Gamma: \frac{x^{2}}{4} - \frac{y^{2}}{5} = 1$ have foci $F_1$ and $F_2$. Let $P$ be a point on the hyperbola in the first quadrant. The centroid of triangle $\triangle P F_1 F_2$ is $G$, and the incenter is $I$.

1. Determine if there exists a point $P$ such that $IG \parallel F_{1} F_{2}$. If so, find the point $P$; if not, explain the reason.
2. Let $A$ be the left vertex of the hyperbola $\Gamma$, and let line $l$ pass through the right focus $F_2$ intersecting the hyperbola $\Gamma$ at points $M$ and $N$. If the slopes of $AM$ and $AN$ are $k_1$ and $k_2$ respectively, and $k_1 + k_2 = -\frac{1}{2}$, find the equation of line $l$.

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