
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
Define the sequences  $x_0,x_1,x_2,\ldots$  and  $y_0,y_1,y_2,\ldots$  such that  $x_0=1$ ,  $y_0=2021$ , and for all nonnegative integers  $n$ , we have  $x_{n+1}=\sqrt{x_ny_n}$  and  $y_{n+1}=\frac{x_n+y_n}{2}.$  There is some constant  $X$  such that as  $n$  grows large,  $x_n-X$  and  $y_n-X$  both approach  $0$ . Estimate  $X$ .
    
An estimate of  $E$  earns  $\max(0,2-0.02|A-E|)$  points, where  $A$  is the actual answer.


*2021 CCA Math Bonanza Lightning Round #5.2*

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