
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 complex number $z$ such that $|z| = \sqrt{2}$, and the imaginary part of $z^2$ is 2.  
(1) Find the complex number $z$;  
(2) Let the points corresponding to $z$, $\left(\overline{z}\right)^2$, and $z - z^2$ in the complex plane be A, B, and C, respectively. Find the area of triangle $ABC$;  
(3) If the point corresponding to the complex number $z$ is in the first quadrant of the complex plane, and the complex number $m$ satisfies $|m-z|=1$, find the maximum and minimum values of $|m|$.

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