
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 a parabola $y^2 = 2px (p > 0)$ with focus $F$, a line $l$ passing through point $F$ at an angle of $\frac{\pi}{4}$ with the x-axis intersects the parabola at points $A$ and $B$. If the circle with diameter $AB$ passes through the point $\left(-\frac{p}{2}, 2\right)$, determine the equation of the parabola.

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