
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
Let the focus of the parabola $C:y^{2}=2px\left(p \gt 0\right)$ be $F$, point $D\left(p,0\right)$, and the line passing through $F$ intersecting $C$ at points $M$ and $N$. When the line $MD$ is perpendicular to the $x$-axis, $|MF|=3$.  
$(1)$ Find the equation of $C$;  
$(2)$ Let the other intersection points of lines $MD$, $ND$ with $C$ be $A$, $B$ respectively. Denote the inclinations of lines $MN$ and $AB$ as $\alpha$ and $\beta$. When $\alpha -\beta$ reaches its maximum value, find the equation of line $AB$.

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