
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. We define the sequence $\alpha_{0}=\alpha, \alpha_{n+1}=\alpha^{\alpha_{n}},$ and so on. When does the limit $\lim _{n \rightarrow \infty} \alpha_{n}$ exist?

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