
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 $C$: $y^{2}=2px(p > 0)$ with focus $F$, and point $D(2,y_{0})$ is on the parabola $C$ and $|DF|=3$. The line $y=x-1$ intersects the parabola $C$ at points $A$ and $B$, and $O$ is the origin.  
$(1)$ Find the equation of the parabola $C$;  
$(2)$ Find the area of $\triangle OAB$.

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