
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 \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^3 + ax^2 + bx \). Suppose there exist three real numbers \( x_1, x_2, x_3 \) such that \( x_1 + 1 \leq x_2 \leq x_3 - 1 \) and \( f(x_1) = f(x_2) = f(x_3) \). Find the minimum value of \( |a| + 2|b| \).

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