
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
In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is given by $\begin{cases}x=1+ \frac{ \sqrt{2}}{2}t \\ y= \frac{ \sqrt{2}}{2}t\end{cases} (t$ is the parameter$)$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $\rho=4\cos \theta$. Line $l$ intersects circle $C$ at points $A$ and $B$.

$(1)$ Find the parametric equation of circle $C$ and the standard equation of line $l$;

$(2)$ Find the area of $\triangle AOB$.

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