
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 \( z_{1} = \frac{z}{1 + z^2} \) and \( z_{2} = \frac{z^2}{1 + z} \) are both real numbers, find \( z \).

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