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

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