
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 natural number \( n \) greater than 1, find all pairs of integers \( a \) and \( b \) such that the two equations \( x^{n} + ax - 2008 = 0 \) and \( x^{n} + bx - 2009 = 0 \) have at least one common real root.

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