
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 that \( n \) is a positive integer and \( A \) is a complex number with \( i^{2}=-1 \), find the necessary and sufficient condition (in terms of \( A \)) for the equation \(\left(\frac{1+i \cdot x}{1-i \cdot x}\right)^{n}=A\) to have all distinct real roots.

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