
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
Let $a$, $b$, $c$ be three distinct real numbers such that the equation $x^2+ax+1=0$ and $x^2+bx+c=0$ have one common real root, and the equation $x^2+x+a=0$ and $x^2+cx+b=0$ also have one common real root. Find the value of $a+b+c$.

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