
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
How many ways are there to divide the set of numbers \(\left\{2^{0}, 2^{1}, 2^{2}, \cdots, 2^{2005}\right\}\) into two non-empty, disjoint subsets \(A\) and \(B\) such that the equation \(x^{2} - S(A)x + S(B) = 0\) has integer roots, where \(S(M)\) denotes the sum of all elements in the set \(M\)?

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