
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 be two natural numbers  $ m,n, $  and  $ m $  pairwise disjoint sets of natural numbers  $ A_0,A_1,\ldots ,A_{m-1}, $  each having  $ n $  elements, such that no element of  $ A_{i\pmod m} $  is divisible by an element of  $ A_{i+1\pmod m} , $  for any natural number  $ i. $  
Determine the number of ordered pairs   $$  (a,b)\in\bigcup_{0\le j < m} A_j\times\bigcup_{0\le j < m} A_j  $$   such that  $ a|b $  and such that  $ \{ a,b \}\not\in A_k, $  for any  $ k\in\{ 0,1,\ldots ,m-1 \} . $ *Radu Bumbăcea*

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