
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  $P(x)=x^3+ax^2+bx+c$  be a polynomial where  $a,b,c$  are integers and  $c$   is odd. Let  $p_{i}$  be the value of  $P(x)$  at  $x=i$ . Given that  $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of  $p_{2}+2p_{1}-3p_{0}.$ 

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