
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
A sequence $\{x_n\}$ is defined as $x_1 = 3$ and $x_{n+1} = \left\lfloor \sqrt{2} x_n \right\rfloor$ for $n \in \mathbf{N}^{*}$. Find all $n$ such that $x_n, x_{n+1}, x_{n+2}$ form an arithmetic sequence. (Here, $\left\lfloor x \right\rfloor$ denotes the greatest integer less than or equal to the real number $x$.)

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