
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 $z = x + yi$ (where $x, y \in \mathbb{R}$) such that $|z| = \sqrt{2}$ and the imaginary part of $z^2$ is 2, with the corresponding point A of $z$ located in the first quadrant.
1. Find $z$.
2. If $z$, $z^2$, and $z - z^2$ correspond to points A, B, C in the complex plane, respectively, find $\cos(\angle ABC)$.

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