
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
Given a cube \( ABCD-EFGH \) with edge length 2, point \( P_1 \) is randomly selected on the inscribed circle of square \( ABFE \), point \( P_2 \) is randomly selected on the inscribed circle of square \( BCFG \), and point \( P_3 \) is randomly selected on the inscribed circle of square \( EFGH \). Find the minimum and maximum values of \( \left|P_{1} P_{2}\right| + \left|P_{2} P_{3}\right| + \left|P_{3} P_{1}\right| \).

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