
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 that  $a=2$  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 positive integers  $(a,b)$  with  $1\le a,b\le 2011$  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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