
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 $T$ be the set of complex numbers of the form $u + vi,$ where $u$ and $v$ are real numbers, such that 
\[\frac{1}{2} \le u \le \frac{\sqrt{2}}{2}.\] Find the smallest positive integer $n$ such that for all positive integers $m \ge n,$ there exists a complex number $z \in T$ such that $z^m = 1.$

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