
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 decagon $B_1B_2B_3B_4B_5B_6B_7B_8B_9B_{10}$ is inscribed in a circle of area $1$ square units. 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 $\tfrac{1}{10},$ while the region bounded by $\overline{QB_3},\overline{QB_4},$ and the minor arc $\widehat{B_3B_4}$ of the circle has area $\tfrac{1}{12}.$ There is a positive integer $m$ such that the area of the region bounded by $\overline{QB_7},\overline{QB_8},$ and the minor arc $\widehat{B_7B_8}$ of the circle is equal to $\tfrac{1}{11}-\tfrac{\sqrt{3}}{m}.$ Find $m.$

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