
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  $\omega$  be the unit circle in the  $xy$ -plane in  $3$ -dimensional space. Find all points  $P$  not on the  $xy$ -plane that satisfy the following condition: There exist points  $A,B,C$  on  $\omega$  such that  $$  \angle APB = \angle APC = \angle BPC = 90^\circ. $$  

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