
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 $x\_1$ and $x\_2$ are the two roots of the quadratic equation $ax^2+bx+c=0$ with real coefficients. If $x\_1$ is a complex number, $\frac{x\_1^2}{x\_2}$ is a real number, then $S=1+\frac{x\_1}{x\_2}+(\frac{x\_1}{x\_2})^2+(\frac{x\_1}{x\_2})^4+(\frac{x\_1}{x\_2})^8+(\frac{x\_1}{x\_2})^{16}+(\frac{x\_1}{x\_2})^{32}=$ \_\_\_\_\_\_.

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