
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  $a_1,a_2,a_3,a_4$  be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let  $i,j,k\in\{1,2,3,4\}$  with  $i \neq j$ ,  $j\neq k$ , and  $k\neq i $ . Determine the maximum number of triples  $(i,j,k)$  for which   $$ ({\rm gcd}(a_i,a_j))^2|a_k. $$  

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