
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 circle $O$ is $\begin{cases} & x=2\cos \theta \\ & y=2\sin \theta \end{cases}$ ($\theta$ is the parameter), and the parametric equation of line $l$ is $\begin{cases} & x=2+t \\ & y=4+t \end{cases}$ ($t$ is the parameter).

$(1)$ If line $l$ intersects circle $O$ at points $A$ and $B$, find the length of chord $|AB|$;

$(2)$ With the origin $O$ of this Cartesian coordinate system as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system. The polar equation of circle $C$ is $\rho =2\cos \theta +2\sqrt{3}\sin \theta$. If circle $O$ and circle $C$ intersect at points $P$ and $Q$, find the equation of the line containing chord $PQ$ in Cartesian coordinates.

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