
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 equations of curve $({C_{1}})$ are given by $\begin{cases} x=1+\cos \alpha , \\ y=\sin \alpha \end{cases}(\alpha \text{ is the parameter})$; in the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of curve $({C_{2}})$ is given by $\rho {\cos ^{2}}\theta =\sin \theta $.

(1) Find the polar equation of curve $({C_{1}})$ and the Cartesian equation of curve $({C_{2}})$;

(2) If ray $l$: $y=kx(x\geqslant 0)$ intersects with curves $({C_{1}})$, $({C_{2}})$ at points $A$, $B$ (respectively, and $A$, $B$ are distinct from the origin), find the range of values of $|OA|\cdot |OB|$ when the slope $k\in (1,\sqrt{3}]$.

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