
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
Which is greater:

$$
\frac{2,00000000004}{1,0 \text{ J)JJJ0jj4 }} \text{ or }\frac{2,00000000002}{(1,00000000002)^{2}+2,00000000002}?
$$

Given \(a > b > 0\), which is greater:

$$
\frac{1 + a + a^{2} + \ldots + a^{n-1}}{1 + a + a^{2} + \ldots + a^{n}} \text{ or }\frac{1 + b + b^{2} + \ldots + b^{n-1}}{1 + b + b^{2} + \ldots + b^{n}}?
$$

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