
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
Together, Kenneth and Ellen pick a real number  $a$ . Kenneth subtracts  $a$  from every thousandth root of unity (that is, the thousand complex numbers  $\omega$  for which  $\omega^{1000}=1$ ) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts  $a$  from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of  $a$  are there?

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