
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
In the Cartesian coordinate plane $(xOy)$, the right branch of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\phantom{\rule{0.33em}{0ex}}(a>0,b>0)$ intersects with the parabola ${x}^{2}=2py\phantom{\rule{0.33em}{0ex}}(p>0)$ at points $A$ and $B$. If $|AF|+|BF|=4|OF|$, find the equations of the asymptotes of the hyperbola.

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