
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  $P(x) = x^2 + ax + b$  be a quadratic polynomial. For how many pairs  $(a, b)$  of positive integers where  $a, b < 1000$  do the quadratics  $P(x+1)$  and  $P(x) + 1$  have at least one root in common?

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