
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
Suppose that \(\omega\) is a primitive \(2007^{\text{th}}\) root of unity. Find \((2^{2007} - 1) \sum_{j=1}^{2000} \frac{1}{2 - \omega^j}\). Express your answer in the form \(m \cdot n^k + p\), where \(m, n, k\), and \(p\) are positive integers. Note that a number \(z\) is a primitive \(n^{\text{th}}\) root of unity if \(z^n = 1\) and \(n\) is the smallest number amongst \(k=1,2, \ldots, n\) such that \(z^k = 1\).

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