
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 the function
$$
f(x) = x^3 + ax^2 + bx + c \quad \text{for } x \in \mathbf{R},
$$
where \( a \), \( b \), and \( c \) are distinct non-zero integers, and
$$
f(a) = a^3 \quad \text{and} \quad f(b) = b^3.
$$
Find \( a + b + c = \quad \).

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