
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 circle $C_{1}: x^{2}+(y-2)^{2}=4$ and parabola $C_{2}: y^{2}=2px (p > 0)$ intersect at points $A$ and $B$, and $|AB|= \frac {8 \sqrt {5}}{5}$. Determine the equation of the parabola $C_{2}$.

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