
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 points $A(-1,2)$, $C(-1,0)$, and the symmetric point of point $A$ with respect to the line $x-y+1=0$ is $B$. <br/> $(1)$ Find the equation of the circumcircle of $\triangle ABC$; <br/> $(2)$ A tangent line to the circumcircle of $\triangle ABC$ passing through point $P(\sqrt{2},2)$ is drawn. Find the equation of the tangent line.

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