
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 decagon \(A_1A_2A_3A_4A_5A_6A_7A_8A_9A_{10}\) is inscribed in a circle of area \(1.\) Point \(P\) is inside the circle so that the region bounded by \(\overline{PA_1},\overline{PA_2},\) and the minor arc \(\widehat{A_1A_2}\) of the circle has area \(\frac{1}{10},\) the region bounded by \(\overline{PA_3},\overline{PA_4},\) and the minor arc \(\widehat{A_3A_4}\) of the circle has area \(\frac{1}{12}.\) Determine the area of the region bounded by \(\overline{PA_7},\overline{PA_8},\) and the minor arc \(\widehat{A_7A_8}\) of the circle.

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