
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  $i$  be a root of the equation  $x^2+1=0$  and let  $\omega$  be a root of the equation  $x^2+x+1=0$  . Construct a polynomial   $$ f(x)=a_0+a_1x+\cdots+a_nx^n $$   where  $a_0,a_1,\cdots,a_n$  are all integers such that  $f(i+\omega)=0$  .

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