
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
Find the necessary and sufficient condition for the complex number \( a + b i \) to be:
1. Equal to its conjugate,
2. Equal to the inverse of its conjugate,
3. Equal to the opposite of its conjugate.

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