
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  $ ABCD$  be a regular tetrahedron, and let  $ O$  be the centroid of triangle  $ BCD$ . Consider the point  $ P$  on  $ AO$  such that  $ P$  minimizes  $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$ . Find  $ \sin \angle PBO$ .

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