
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

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

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

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