
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 $x + yi$, where $x$ and $y$ are real numbers, such that
\[
\frac{1}{2} \leq x \leq \frac{\sqrt{3}}{2} \quad \text{and} \quad \frac{\sqrt{2}}{2} \leq y \leq 1.
\]
Find the smallest positive integer $m$ such that for all positive integers $n \geq m$, there exists a complex number $z \in T$ such that $z^n = i$, where $i$ is the imaginary unit.

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