
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
Given the hyperbola $\Gamma$: $x^{2}- \frac{y^{2}}{8}=1$ with left and right foci $F\_1$ and $F\_2$, respectively. Let $A$ be the left vertex of the hyperbola $\Gamma$, and the line $l$ passes 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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