
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
Given that circle $C\_1$: $x^{2}+y^{2}+Dx+Ey+F=0$ is symmetric about the line $x+y-2=0$, and passes through the points $(0,0)$ and $(4,0)$.
(I) Find the standard equation of circle $C\_1$;
(II) Given that the equation of circle $C\_2$ is $(x-2)^{2}+y^{2}=1$.
(i) If the length of the chord intercepted by the line $l$ passing through the origin and circle $C\_2$ is $ \sqrt {2}$, find the equation of $l$;
(ii) Given that the line $m$ with slope $k$ passes through a moving point on circle $C\_2$, and intersects circle $C\_1$ at points $A$ and $B$, and the ray $PC\_2$ intersects circle $C\_1$ at point $Q$, find the maximum area of $ \triangle ABQ$.

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