
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  $n$  and  $m$  be natural numbers such that  $m+ i=a_ib_i^2$  for  $i=1,2, \cdots n$  where  $a_i$  and  $b_i$  are natural numbers and  $a_i$  is not divisible by a square of a prime number.
Find all  $n$  for which there exists an  $m$  such that  $\sum_{i=1}^{n}a_i=12$ 

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