
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 one of the foci of a hyperbola is $(4,0)$, and its eccentricity $e=2$.  
$(1)$ Find the standard equation of the hyperbola;  
$(2)$ Write the equation of the asymptotes of the hyperbola, and calculate the distance from its focus $(4,0)$ to the other asymptote.

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