
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 the center of circle $C$ is at coordinates $(1,1)$, and line $l:x+y=1$ intercepts a chord of length $\sqrt{2}$ on circle $C$.

(I) Find the equation of circle $C$;

(II) From a point $P(2,3)$ outside circle $C$, draw a tangent line to the circle, and find the equation of the tangent line.

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