
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
The line $l$ passing through point $A(2,0)$ intersects the parabola $C: y^2 = 2px (p > 0)$ at points $M$ and $N$ (where $M$ is in the first quadrant). When the slope of the line $l$ is $\frac{π}{4}$, $|MN| = 3\sqrt{2}$. Find:<br/>
$(1)$ The equation of the parabola;<br/>
$(2)$ If $B(3,0)$, the extension of $MB$ intersects the parabola $C$ at point $P$, and the extension of $PN$ intersects the $x$-axis at point $Q$, find the value of $\frac{|QN|}{|QP|}$.

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