
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 that a line $l$ with slope $k$ passes through the point $P(3,0)$ and intersects the right branch of the hyperbola $C: x^{2}-\frac{y^{2}}{3}=1$ at points $A$ and $B$. The point $F$ is the right focus of the hyperbola $C$, and it is known that $|AF| + |BF| = 16$. Find the value of $k$.

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