
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 $y^{2}=2px (p > 0)$ and its focus, a line with a slope of $2 \sqrt {2}$ intersects the parabola at points $A(x_{1},y_{1})$ and $B(x_{2},y_{2}) (x_{1} < x_{2})$, with $|AB|=\frac{9}{2}$.

1. Find the equation of the parabola.
2. Through the parabolic point $M(1, 2)$, draw two perpendicular lines $MP$ and $MQ$ that intersect the parabola at points $P$ and $Q$, respectively. Is the line $PQ$ passing through a fixed point? If so, find it; otherwise, explain why not.
3. Find the maximum distance between the origin $O$ and the line $PQ$.

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