
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
For positive integers \(a\) and \(b\), let \(M(a, b)=\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}\), and for each positive integer \(n \geq 2\), define
\[
x_{n}=M(1, M(2, M(3, \ldots, M(n-2, M(n-1, n)) \ldots))).
\]

Compute the number of positive integers \(n\) such that \(2 \leq n \leq 2021\) and \(5 x_{n}^{2}+5 x_{n+1}^{2}=26 x_{n} x_{n+1}\).

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