
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 rectangular coordinate system $xOy$, the parametric equations of the curve $C_1$ are given by $$\begin{cases} x=1+ \sqrt {3}\cos\alpha \\ y= \sqrt {3}\sin\alpha\end{cases}$$ (where $\alpha$ is the parameter), and the polar coordinate equation of the curve $C_2$ with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis is $\rho=2\sin\theta$.
(1) Find the Cartesian equation of curve $C_1$ and the polar coordinate equation of curve $C_2$;
(2) Determine whether curves $C_1$ and $C_2$ have two intersection points. If they do, find the distance between the two intersection points; otherwise, explain why not.

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