
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 \(x^{2} + a x + b\) and \(x^{2} + b x + c\) is \(x + 1\) (in the ring of polynomials in \(x\) with integer coefficients), and the least common multiple of \(x^{2} + a x + b\) and \(x^{2} + b x + c\) is \(x^{3} - 4 x^{2} + x + 6\). Find \(a + b + c\).

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