
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
a) Factor the expression

$$
a^{3}+b^{3}+c^{3}-3abc
$$

b) Using the result from part a), find the general formula for solving the cubic equation

$$
x^{3}+px+q=0
$$

Note: The result from this problem allows solving any cubic equation. For example, consider the cubic equation

$$
x^{3}+Ax^{2}+Bx+C=0
$$

By making the substitution \( x = y + c \), we get:

$$
y^{3}+3cy^{2}+3c^{2}y+c^{3}+A(y^{2}+2cy+c^{2})+B(y+c)+C=0
$$

From which, by setting \( c = -\frac{A}{3} \) (i.e. \( x = y - \frac{A}{3} \)), we arrive at the equation

$$
y^{3} + \left( \frac{3A^{2}}{9} - \frac{2A^{2}}{3} + B \right) y + \left( -\frac{A^{3}}{27} + \frac{A^{3}}{9} - \frac{AB}{3} + C \right) = 0
$$

which is in the form:

$$
y^{3}+py+q=0, \quad \text{where} \quad p=-\frac{A^{2}}{3}+B, \quad q=\frac{2A^{3}}{27}-\frac{AB}{3}+C
$$

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