
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
Regular hexagon \(B_1B_2B_3B_4B_5B_6\) is inscribed in a circle of unit area. Point \(Q\) lies inside the circle such that the region bounded by \(\overline{QB_1},\overline{QB_2},\) and the minor arc \(\widehat{B_1B_2}\) of the circle has area \(\frac{1}{12},\) and the region bounded by \(\overline{QB_3},\overline{QB_4},\) and the minor arc \(\widehat{B_3B_4}\) has area \(\frac{1}{15}.\) There is a positive integer \(m\) such that the area of the region bounded by \(\overline{QB_5},\overline{QB_6},\) and the minor arc \(\widehat{B_5B_6}\) of the circle is equal to \(\frac{1}{18}-\frac{\sqrt{3}}{m}.\) Find \(m.\)

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