
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
Suppose \( a, b \) and \( c \) are integers such that the greatest common divisor of \( x^2 + ax + b \) and \( x^2 + bx + c \) is \( x + 1 \) (in the set of polynomials in \( x \) with integer coefficients), and the least common multiple of \( x^2 + ax + b \) and \( x^2 + bx + c \) is \( x^3 - 4x^2 + x + 6 \). Find \( a + b + c \).

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