
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 parabola $C: y^2 = 2px (p > 0)$ with a focus point $F$, let $M(x_0, 2\sqrt{2})$ be a point on parabola $C$. A circle centered at $M$ with a radius $|MF|$ intersects the $y$-axis, producing a chord with a length of $2\sqrt{5}$. Find the value of $p$.

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