
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 $\alpha,$ $\beta,$ $\gamma$ be the roots of the cubic $x^3 - 4x^2 + 6x + 8 = 0.$ Given that $P(x)$ is a cubic polynomial such that $P(\alpha) = \beta + \gamma,$ $P(\beta) = \alpha + \gamma,$ $P(\gamma) = \alpha + \beta,$ and $P(\alpha + \beta + \gamma) = -20,$ find $P(x).$

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