
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 $t \in \mathbb{R}$, and the circle $C: x^2 + y^2 - 2tx - 2t^2y + 4t - 4 = 0$.

(1) If the center of circle $C$ lies on the line $x - y + 2 = 0$, find the equation of circle $C$;  
(2) Does circle $C$ pass through a fixed point? If it does, find the coordinates of the fixed point; if not, explain why.

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