
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  $ n, m$  be positive integers of different parity, and  $ n > m$ .  Find all integers  $ x$  such that  $ \frac {x^{2^n} \minus{} 1}{x^{2^m} \minus{} 1}$  is a perfect square.

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