
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 \(\frac{S_{1}}{S_{2}}\), where

\[ S_{1} = \frac{1}{2^{2019}}+\frac{1}{2^{2018}}-\frac{1}{2^{2017}}+\ldots+\frac{1}{2^{3}}+\frac{1}{2^{2}}-\frac{1}{2}, \]
\[
S_{2} = \frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{2^{3}}+\ldots+\frac{1}{2^{2017}}+\frac{1}{2^{2018}}-\frac{1}{2^{2019}}
\]

(in both sums, the signs of the terms alternate as \(++-++-++- \ldots\)).

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