
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 \(\left(x_{n}\right), n \geq 0\), is defined as follows: \(x_{0}=a, x_{1}=2\) and \(x_{n}=2 x_{n-1} x_{n-2} - x_{n-1} - x_{n-2} + 1\) for \(n > 1\). Find all integers \(a\) such that \(2 x_{3n} - 1\) is a perfect square for all \(n \geq 1\).

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