
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 a circle is 
$$
\begin{cases} 
x= \sqrt {2}\cosθ \\ 
y= \sqrt {2}\sinθ
\end{cases}
$$
($θ$ is the parameter), and the parametric equation of line $C_1$ is 
$$
\begin{cases} 
x=1+t \\ 
y=2+t
\end{cases}
$$
($t$ is the parameter).
(1) If line $C_1$ intersects circle $O$ at points $A$ and $B$, find the length of chord $|AB|$;
(2) Taking the origin $O$ of the Cartesian coordinate system as the pole and the non-negative half-axis of the $x$-axis as the polar axis to establish a polar coordinate system, the polar equation of circle $C_2$ is $$ρ=2\cosθ+2\sqrt{3}\sinθ$$, and the intersection points of circle $O$ and circle $C_2$ are $P$ 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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