
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 focus $F$, and a circle centered at point $M(x_0, 2\sqrt{2})$ ($x_0 > \frac{p}{2}$) on the parabola $C$ intersects with the line segment $MF$ at point $A$, and the chord cut by the line $x = \frac{p}{2}$ has a length of $\sqrt{3}|\overrightarrow{MA}|$. If $\frac{|\overrightarrow{MA}|}{|\overrightarrow{AF}|} = 2$, then $|\overrightarrow{AF}| = \_\_\_\_\_\_$.

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