
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
Given the function $g(x) = ax^3 + bx^2 + cx + d$ $(a \neq 0)$, its derivative is $f(x)$, and $a + 2b + 3c = 0$, $f(0)f(1) > 0$. Let $x_1$, $x_2$ be the two roots of the equation $f(x) = 0$, calculate the range of $|x_1 - x_2|$.

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