
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 the curve $C_1$ are given by $\begin{cases} x=2\cos\alpha \\ y=2+2\sin\alpha \end{cases}$ (where $\alpha$ is the parameter). In the polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar equation of the curve $C_2$ is given by $\rho\sin^2\theta=\cos\theta$.

1. Find the polar equation of the curve $C_1$ and the Cartesian equation of the curve $C_2$.
2. If the ray $l: \theta=\alpha$ ($\rho \geq 0$, $\alpha \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$) intersects the curves $C_1$ and $C_2$ at points $A$ and $B$ respectively (where $A$ and $B$ are distinct from the origin), find the range of values for $|OA|\cdot|OB|$.

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