
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
A regular dodecagon \( B_1B_2B_3\ldots B_{12} \) is inscribed in a circle of area \( 4\pi \). A point \( Q \) inside the circle is such that the region bounded by \( \overline{QB_1}, \overline{QB_2} \), and the minor arc \( \hat{B_1B_2} \) of the circle has area \( \frac{1}{8} \), while the region bounded by \( \overline{QB_4}, \overline{QB_5} \), and the minor arc \( \hat{B_4B_5} \) of the circle has area \( \frac{1}{10} \). There is a positive integer \( m \) such that the area of the region bounded by \( \overline{QB_9},\overline{QB_{10}} \), and the minor arc \( \hat{B_9B_{10}} \) of the circle is equal to \( \frac{4}{15} - \frac{\sqrt{3}}{m} \). Find \( m \).

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