
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
Elective 4-4: Coordinate System and Parametric Equations

In the plane Cartesian coordinate system $xOy$, the equation of curve $C$ is $x^2-2x+y^2=0$. Taking the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of line $l$ is $\theta =\frac{\pi }{4} (\rho \in \mathbb{R})$.

$(1)$ Write the polar equation of $C$ and find the polar coordinates of the intersection points $M, N$ of $l$ and $C$;

$(2)$ Let $P$ be a moving point on the ellipse $\frac{x^2}{3}+y^2=1$, find the maximum area of $\Delta PMN$.

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