
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 \(\frac{1}{2} \le x \le \frac{2}{3}\). Find the smallest positive integer \(m\) such that for all positive integers \(n \ge m\), there exists a complex number \(z \in S\) such that \(z^n = 1\).

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