
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
For integers  $a$  and  $b$  with  $0 \leq a,b < {2010}^{18}$  let  $S$  be the set of all polynomials in the form of  $P(x)=ax^2+bx.$  For a polynomial  $P$  in  $S,$  if for all integers n with  $0 \leq n <{2010}^{18}$  there exists a polynomial  $Q$  in  $S$  satisfying  $Q(P(n)) \equiv n \pmod {2010^{18}},$  then we call  $P$  as a *good polynomial.*
Find the number of *good polynomials.*

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