
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 a line $l$ intersects with circle $C$ at points $P(1,0)$ and $Q(0,1)$.
(1) Find the equation of the line where the center $C$ of the circle is located;
(2) If the radius of the circle center $C$ is 1, find the equation of circle $C$.

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