
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 of curve $C_1$ is $$\begin{cases} x=1+\cos\alpha \\ y=\sin\alpha\end{cases}$$ (where $\alpha$ is the parameter); in the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho\cos^2\theta=2\sin\theta$;  
(1) Find the polar equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;  
(2) If the ray $l: y=kx$ ($x\geq0$) intersects curves $C_1$ and $C_2$ at points $A$ and $B$ respectively (where $A$ and $B$ are different from the origin), and when the slope $k\in[1, \sqrt{3})$, find the range of values for $|OA|\cdot|OB|$.

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