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

(Ⅰ) Find the general equation of $C_{1}$ and the Cartesian coordinate equation of $C_{2}$;

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

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