
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
The asymptotic line equations of the hyperbola $C$: $\frac{x^{2}}{16} - \frac{y^{2}}{4} = 1$ are $\_\_\_\_\_\_$; Let $F\_1$, $F\_2$ be the left and right foci of the hyperbola $C$, and $P$ be a point on $C$ such that $|PF\_1| = 4$, then $|PF\_2| =$ $\_\_\_\_\_\_$.

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