
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
Let  $PA_1A_2...A_{12} $   be the regular pyramid,  $ A_1A_2...A_{12} $  is regular polygon,   $S$  is area of the triangle  $PA_1A_5$  and angle  between of the planes  $A_1A_2...A_{12} $  and  $ PA_1A_5 $  is equal to  $ \alpha $ .
Find the volume of the pyramid.

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