
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
Given that the positive sequence $\{a_n\}$ is a geometric sequence with a common ratio not equal to $1$, and $\lg a_1 + \lg a_{2019} = 0$, calculate the sum $f(a_1) + f(a_2) + \ldots + f(a_{2019})$ where $f(x) = \frac{2}{1+x^2}$.

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