
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
A chord $AB$ that makes an angle of $\frac{\pi}{6}$ with the horizontal passes through the left focus $F_1$ of the hyperbola $x^{2}- \frac{y^{2}}{3}=1$. Find:  
$(1)$ The length of the segment $AB$;  
$(2)$ Let $F_{2}$ be the right focus, find the perimeter of $\triangle F_{2}AB$.

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