
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
Elective 4-4: Coordinate System and Parametric Equations.

In the Cartesian coordinate system $xOy$, the parametric equation of line $L$ is $\begin{cases}x=2+t\cos \alpha \\ y=t\sin \alpha\end{cases} (t$ 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 ${p}^{2}{\cos }^{2}\theta+2{p}^{2}{\sin }^{2}\theta=12$, and line $l$ intersects curve $C$ at points $P$ and $Q$

$(1)$ Find the general 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 $\left|AP\right|\left|AQ\right|=6$, find the general equation of line $L$

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