
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 $z=x+yi (x,y∈R)$ that satisfies $|z|=\sqrt{2}$, the imaginary part of $z^2$ is $2$, and point $A$ corresponding to $z$ is in the first quadrant.

1. Find $z$.
2. If $z$, $z^2$, and $z-z^2$ correspond to points $A$, $B$, and $C$ in the complex plane, find $\cos∠ABC$.

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