
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 tetrahedron  $P_1P_2P_3P_4$  has side length  $1$ . Define  $P_i$  for  $i > 4$  to be the centroid of tetrahedron  $P_{i-1}P_{i-2}P_{i-3}P_{i-4}$ , and  $P_{ \infty} = \lim_{n\to \infty} P_n$ . What is the length of  $P_5P_{ \infty}$ ?

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