
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 sets of positive integers  $\{x_1, x_2, \dots, x_{20}\}$  such that   $$ x_{i+2}^2=lcm(x_{i+1}, x_{i})+lcm(x_{i}, x_{i-1}) $$   for  $i=1, 2, \dots, 20$  where  $x_0=x_{20}, x_{21}=x_1, x_{22}=x_2$ .

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