
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 as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The polar equation of curve $C_{1}$ is $\rho^{2}-4\rho\cos \theta+3=0$, where $\theta\in[0,2\pi]$.
$(1)$ Find the Cartesian equation of $C_{1}$;
$(2)$ The parametric equation of curve $C_{2}$ is $\begin{cases} x=t\cos \frac {\pi}{6} \\ y=t\sin \frac {\pi}{6}\end{cases}$ (where $t$ is the parameter), find the polar coordinates of the intersection points of $C_{1}$ and $C_{2}$.

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