
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 $y^{2}=2px\left(p \gt 0\right)$ with focus $F$, a line passing through point $F$ intersects the parabola at points $A$ and $B$, where $|{AB}|=5\sqrt{2}$, and the midpoint of $AB$ has a $y$-coordinate of $\sqrt{2}$. Find the value of $p$.

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