
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
Find all positive integers  $n$  such that it is possible to split the numbers from  $1$  to  $2n$  in two groups  $(a_1,a_2,..,a_n)$ ,  $(b_1,b_2,...,b_n)$  in such a way that  $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$ .

*Proposed by Alef Pineda*

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