
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 $C: y^{2}=2px(p$ is a constant, $p \gt 0)$ and the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ with the focus $F$ coinciding with the right focus of the ellipse. The line passing through point $F$ intersects the parabola at points $A$ and $B$. <br/>$(1)$ Find the standard equation of the parabola $C$;<br/>$(2)$ If the slope of line $AB$ is $1$, find $|AB|$.

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