
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
Determine all the positive real numbers  $x_1, x_2, x_3, \dots, x_{2021}$  such that  $x_{i+1}=\frac{x_i^3+2}{3x_i^2}$ for every  $i=1, 2, 3, \dots, 2020$  and  $x_{2021}=x_1$ 

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