
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 $x$ be a complex number such that $x^{3013} = 1$ and $x \neq 1$. Compute the sum
\[
\sum_{k=1}^{1004} \left( \frac{x^{3k}}{x^k - 1} + \frac{x^{6k}}{x^{2k} - 1} \right).
\]

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