
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 that the positive integers \( a_{1}, a_{2}, a_{3}, a_{4} \) satisfy the condition that in any circular arrangement of these four numbers, there exist two adjacent elements which are not coprime. Find the maximum number of ordered triplets \((i, j, k)\) such that \( i, j, k \in {1, 2, 3, 4} \), \( i \neq j \), \( j \neq k \), \( k \neq i \), and \( (a_{i}, a_{j})^2 \mid a_{k} \).

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