
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 a line \( l \) passing through point \( P(0,1) \) with slope \( k \), it intersects the hyperbola \( C: x^{2}-\frac{y^{2}}{3}=1 \) at points \( A \) and \( B \).
1. Find the range of values for \( k \).
2. Let \( F_{2} \) be the right focus of the hyperbola \( C \). Given that \( \left|A F_{2}\right|+\left|B F_{2}\right|=6 \), find the value of \( k \).

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