
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
For a real number  $a$  and an integer  $n(\geq 2)$ , define  $$ S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} $$  Find every value of  $a$  s.t. sequence  $\{S_n(a)\}_{n\geq 2}$  converges to a positive real.

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