
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 \(x_{1}, x_{2}\) be the roots of the quadratic equation \(ax^{2} + bx + c = 0\) with real coefficients. If \(x_{1}\) is an imaginary number and \(\frac{x_{1}^{2}}{x_{2}}\) is a real number, determine the value of \(S = 1 + \frac{x_{1}}{x_{2}} + \left(\frac{x_{1}}{x_{2}}\right)^{2} + \left(\frac{x_{1}}{x_{2}}\right)^{4} + \cdots + \left(\frac{x_{1}}{x_{2}}\right)^{1999}\).

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