
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
Consider sequences of positive real numbers of the form \(x, 3000, y, \dots\) in which every term after the first is \(2\) less than the double of the product of its two immediate neighbors. For how many different values of \(x\) does the term \(3001\) appear somewhere in the sequence?

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