
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 for circle $C$ is 
$$
\begin{cases}
x = 2(1 + \cos \alpha) \\
y = 2\sin \alpha
\end{cases}
\text{(}\alpha\text{ as parameter)}
$$
Set up the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, and let the polar equation of line $l$ be $\theta=\theta_{0}$, with $\theta_{0}\in\left(0, \dfrac{\pi}{2}\right)$ and $\tan \theta_{0} = \dfrac{\sqrt{7}}{3}$.  
(1) Find the polar equation of circle $C$.  
(2) Let $M$ be the intersection point between the line $l$ and circle $C$ in the first quadrant, and find $|OM|$.

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