
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
1. Given a non-zero complex number $z$ that satisfies $|z+2|=2$ and $z+ \frac {4}{z} \in \mathbb{R}$, find the complex number $z$.
2. Given an imaginary number $z$ that makes both $\frac {z^{2}}{z+1}$ and $\frac {z}{z^{2}+1}$ real numbers, find the imaginary number $z$.

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