
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 the hyperbola $C_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) with an eccentricity of $2$, and if the distance from the focus of the parabola $C_2$: $x^2 = 2py$ ($p > 0$) to the asymptote of the hyperbola $C_1$ is $2$, then the equation of the parabola $C_2$ is __________.

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