
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
Given a non-zero real number sequence \( x_{1}, x_{2}, x_{3}, \cdots \) that satisfies
\[ 
x_{n}=\frac{x_{n-2} x_{n-1}}{2 x_{n-2}-x_{n-1}}, \quad n=3,4,5, \cdots 
\]
find the necessary and sufficient condition for \( x_{1} \) and \( x_{2} \) such that the sequence contains infinitely many integers.

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