
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 a straight line $l$ with a parametric equation $\begin{cases} x = 2 + \sqrt{2}t \\ y = \sqrt{2}t \end{cases} (t$ is the parameter$)$, and a circle $C$ with the equation $x^2 + y^2 = 2x$. The line $l$ intersects the circle $C$ at points $A$ and $B$.
$(I)$ Establish a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, and find the polar equations of the straight line $l$ and the circle $C$.
$(II)$ Calculate the area of $\triangle OAB$ (where $O$ is the origin of the coordinates).

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