
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 the line $y=x+b$ intersects with the parabola $y^2=2px$ $(p>0)$ at points A and B. If $\overrightarrow{OA} \perp \overrightarrow{OB}$ (where O is the origin) and the area of triangle $\triangle AOB$ is $2\sqrt{5}$, find the equation of the parabola.

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