
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 with focus $F$, defined by $y^2 = 2px$ ($p > 0$), there is a point $A(m, 2\sqrt{2})$ on the parabola. A circle centered at point A with radius $|AF|$ intersects the y-axis, creating a chord of length $2\sqrt{7}$. Find the value of $m$.

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