
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  $x^2+bx+c$   $x^3-4x^2+x+6$ .  Find  $a+b+c$ .

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