
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 $a,$ $b,$ $c$ be the roots of the cubic $x^3 - 2x^2 - 4x - 1 = 0.$ Given that $P(x)$ is a cubic polynomial such that:
- $P(a) = b + 2c,$
- $P(b) = 2a + c,$
- $P(c) = a + 2b,$
- $P(a + b + c) = -20,$

find $P(x).$

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