
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
 $\omega$  is a complex number such that  $\omega^{2013} = 1$  and  $\omega^m \neq 1$  for  $m=1,2,\ldots,2012$ . Find the number of ordered pairs of integers  $(a,b)$  with  $1 \le a, b \le 2013$  such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer coefficients and leading coefficient  $1$ . (Such complex numbers are called *algebraic integers*.)

*Victor Wang*

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