
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 $x^3+ax^2+bx+c$ and $x^3+bx^2+cx+a$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^3+ax^2+bx+c$ and $x^3+bx^2+cx+a$ is $x^4-3x^3-2x^2+6x+9$.  Find $a+b+c$.

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