
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
Let \( S \) be the set of complex numbers of the form \( x + yi \), where \( x \) and \( y \) are real numbers, such that
\[
\frac{1}{2} \leq x \leq \frac{\sqrt{3}}{2}.
\]
Find the smallest positive integer \( m \) such that for all positive integers \( n \geq m \), there exists a complex number \( z \in S \) such that \( z^n = 1 \).

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