
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
For a nonnegative integer  $k$ , let  $f(k)$  be the number of ones in the base 3 representation of  $k$ . Find all complex numbers  $z$  such that  $$ \sum_{k=0}^{3^{1010}-1}(-2)^{f(k)}(z+k)^{2023}=0 $$  

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