
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
The sequence of nonzero real numbers \( x_1, x_2, x_3, \ldots \) satisfies \( x_n = \frac{x_{n-2} x_{n-1}}{2x_{n-2} - x_{n-1}} \) for all \( n > 2 \). For which \((x_1, x_2)\) does the sequence contain infinitely many integral terms?

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