
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 $a$, $b$, $c$, and $d$ be the roots of the quartic $x^4 + 4x^3 + 6x^2 + 8x + 10 = 0$. Given that $P(x)$ is a polynomial such that $P(a) = b + c + d$, $P(b) = a + c + d$, $P(c) = a + b + d$, $P(d) = a + b + c$, and $P(a+b+c+d) = -20$, find $P(x)$.

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