
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 \( f(x) = x^2 - ax + b \), where \( a \) and \( b \) are positive integers.

(a) Suppose \( a = 2 \) and \( b = 2 \). Determine the set of real roots of \( f(x) - x \), and the set of real roots of \( f(f(x)) - x \).

(b) Determine the number of pairs of positive integers \( (a, b) \) with \( 1 \leq a, b \leq 2011 \) for which every root of \( f(f(x)) - x \) is an integer.

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