
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
a) Find the value of the sum

$$
\frac{1}{1+1 / x}+\frac{1}{1+x}
$$

b) Find the value of the sum

$$
\frac{1}{2019^{-2019}+1}+\ldots+\frac{1}{2019^{-1}+1}+\frac{1}{2019^{0}+1}+\frac{1}{2019^{1}+1}+\ldots+\frac{1}{2019^{2019}+1}
$$

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