
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 circle C with its center at (3, 1), and circle C is tangent to the line $y=x$.  
(1) The equation of circle C is \_\_\_\_\_\_ ;  
(2) If circle C intersects line $l: x-y+a=0$ ($a \neq 0$) at points A and B, and $|AB|=2$, find the value of $a$.

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