
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  $x^3 + 3x^2 + 5x + 7$ . Given that  $P$  is a cubic polynomial such that  $P(a)=b+c$ ,  $P(b) = c+a$ ,  $P(c) = a+b$ , and  $P(a+b+c) = -16$ , find  $P(0)$ .

*Author: Alex Zhu*

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