
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
 $PA_1A_2...A_n$  is a pyramid. The base  $A_1A_2...A_n$  is a regular n-gon. The apex  $P$  is placed so that the lines  $PA_i$  all make an angle  $60^{\cdot}$  with the plane of the base. For which  $n$  is it possible to find  $B_i$  on  $PA_i$  for  $i = 2, 3, ... , n$  such that  $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$ ?

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