
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
Find the ratio \(\frac{S_1}{S_2}\), where

\[ S_1 = \frac{1}{2^{18}} + \frac{1}{2^{17}} - \frac{1}{2^{16}} + \ldots + \frac{1}{2^3} + \frac{1}{2^2} - \frac{1}{2}, \]

\[ S_2 = \frac{1}{2} + \frac{1}{2^2} - \frac{1}{2^3} + \ldots + \frac{1}{2^{16}} + \frac{1}{2^{17}} - \frac{1}{2^{18}}, \]

with the signs of the terms alternating as \(++-++-++-\ldots\).

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