
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  $f\left(x,y\right)=x^2\left(\left(x+2y\right)^2-y^2+x-1\right)$ . If  $f\left(a,b+c\right)=f\left(b,c+a\right)=f\left(c,a+b\right)$  for distinct numbers  $a,b,c$ , what are all possible values of  $a+b+c$ ?

*2019 CCA Math Bonanza Individual Round #12*

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