
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 $x^7 = 1$ and $x \neq 1$. Compute the sum
\[
\frac{x^2}{x-1} + \frac{x^4}{x^2 - 1} + \frac{x^6}{x^3 - 1} + \frac{x^8}{x^4 - 1} + \frac{x^{10}}{x^5 - 1} + \frac{x^{12}}{x^6 - 1}.
\]

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