
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 $Q$  
$(1)$ Find the Cartesian equation of curve $C$ and the coordinates of the fixed point $A$ through which line $L$ always passes;  
$(2)$ Under the condition of $(1)$, if $|AP||AQ|=6$, find the Cartesian equation of line $L$.

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