
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 area $1$. Point $Q$ lies inside the circle so that the region bounded by $\overline{QB_1},\overline{QB_2},$ and the minor arc $\widehat{B_1B_2}$ of the circle has area $\tfrac{1}{6},$ while the region bounded by $\overline{QB_4},\overline{QB_5},$ and the minor arc $\widehat{B_4B_5}$ of the circle has area $\tfrac{1}{10}.$ There is a positive integer $m$ such that the area of the region bounded by $\overline{QB_3},\overline{QB_4},$ and the minor arc $\widehat{B_3B_4}$ of the circle is equal to $\tfrac{1}{7}-\tfrac{\sqrt{3}}{m}.$ Find $m$.

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