
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 $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the parametric equation of curve $C_1$ is given by $\begin{cases}x=2+\cos \theta \\ y=\sin \theta\end{cases}$ (where $\theta$ is the parameter).

$(1)$ Find the Cartesian coordinate equation of curve $C_1$;

$(2)$ The polar coordinate equation of curve $C_2$ is $\theta= \frac{\pi}{6}$ (where $p \in \mathbb{R}$). Find the polar coordinates of the intersection point(s) of $C_1$ and $C_2$.

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