
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
A circle with center $C$ passes through points $A(0,2)$ and $B(2,0)$, and the center $C$ is on the line $l_{1}$: $2x-y-4=0$.
(I) Find the equation of circle $C$;
(II) Find the length of the chord intercepted by the line $l_{2}$: $3x+4y-8=0$.

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