
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
Let  $x_1,x_2,x_3, \dots$  be a sequence of nonzero real numbers satisfying   $$ x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}} \text{ for } n=3,4,5, \dots. $$   Establish necessary and sufficient conditions on  $x_1$  and  $x_2$  for  $x_n$  to be an integer for infinitely many values of  $n.$ 

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