
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
Let  $\alpha > 0$  be a real number. Compute the limit of the sequence  $\{x_n\}_{n\geq 1}$  defined by   $$ x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha} 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases} $$  

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