
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 line $l$ is $\begin{cases} x=2+t\cos \alpha \\ y=t\sin \alpha \\ \end{cases}$ (where $t$ is the parameter), with $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is $\rho^2\cos^2\theta +2\rho^2\sin^2\theta =12$, and line $l$ intersects curve $C$ at points $P$ and $Q$. The Cartesian equation of curve $C$ is \_\_\_\_\_\_\_. Under condition (1), if $|AP||AQ|=6$, the Cartesian equation of line $l$ is \_\_\_\_\_\_\_.

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