
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
A set of natural numbers is said to be "parfumé" (fragranced) if it contains at least two elements and each of its elements shares a prime factor with at least one of the other elements. Let \( P(n) = n^2 + n + 1 \). Determine the smallest positive integer \( b \) for which there exists a positive integer \( a \) such that the set

\[
\{P(a+1), P(a+2), \ldots, P(a+b)\}
\]

is parfumé.

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