
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 $y^{2}=2px\left(p \gt 0\right)$ with focus $F$ and a line with slope $\sqrt{2}$ intersecting the parabola at points $A(x_{1}$,$y_{1})$ and $B(x_{2}$,$y_{2})$ where $x_{1} \lt x_{2}$ and $|AB|=6$.<br/>$(1)$ Find the equation of the parabola;<br/>$(2)$ The directrix of the parabola intersects the $x$-axis at point $F'$, and a line $l$ passing through point $F'$ intersects the parabola $C$ at points $M$ and $N$. When $\overrightarrow{{F'}M}•\overrightarrow{{F'}N}=12$, find the equation of the line $l$.

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