
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
Find, with proof, a polynomial  $f(x,y,z)$  in three variables, with integer coefficients, such that for all  $a,b,c$  the sign of  $f(a,b,c)$  (that is, positive, negative, or zero) is the same as the sign of  $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ .

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