
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 curve $C$ is given by: 
$$
\begin{cases} 
x= \sqrt {3}\cos \alpha \\ 
y=\sin \alpha
\end{cases}
(\alpha \text{ is a parameter})
$$
In the polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of curve $C_1$ is $\rho=2(0\leqslant \theta\leqslant \pi)$, and the polar equation of ray $l$ is $\theta=a_0(a_0\in[0, \frac {\pi}{2}])$  
$(1)$ Write the polar equation of curve $C$ and the Cartesian coordinate equation of curve $C_1$;  
$(2)$ If ray $l$ intersects curves $C_1$ and $C$ at points $A$ and $B$ respectively, find the range of values for $|AB|$.

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