
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 that the focus of the parabola $y^{2}=2px\left(p \gt 0\right)$ is $F\left(4,0\right)$, and $O$ is the origin.  
$(1)$ Find the equation of the parabola.  
$(2)$ A line with a slope of $1$ passes through point $F$ and intersects the parabola at points $A$ and $B$. Find the area of $\triangle AOB$.

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