
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 $x^{2027}=1$ and $x\neq 1$. Compute the sum:
\[
\frac{x^3}{x-1} + \frac{x^9}{x^2-1} + \frac{x^{18}}{x^4-1} + \dots + \frac{x^{3 \cdot 2025}}{x^{2025}-1}
\]
where the exponents in the numerators are triple the previous one, starting from 3, and the exponents in the denominators follow the sequence $1, 2, 4, \ldots, 2025$ doubling each previous one.

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