
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  $n\geqslant 2$  such that there exists a permutation  $a_1$ ,  $a_2$ ,  $a_3$ , \ldots,  $a_{2n}$  of the numbers   $1, 2, 3, \ldots, 2n$  satisfying   $$ a_1\cdot a_2 + a_3\cdot a_4 + \ldots + a_{2n-3} \cdot  a_{2n-2} = a_{2n-1} \cdot a_{2n}. $$  

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