
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 positive integers is called fragrant if it has at least 2 elements, and for every element in the set, there is at least one other element in the set such that the two elements have a common prime divisor. Let \( P(n) = n^2 + n + 1 \). What is the smallest positive integer \( b \) for which there exists a non-negative integer \( a \) such that the set

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

is fragrant?

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