
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 $l$: $x+y=1$ is intercepted by circle $C$ with a chord length of $ \sqrt {2}$,
$(1)$ 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 tangent line's equation.

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