
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
The sequence $\left\{a_{n}\right\}$ satisfies $a_{0} = 0, a_{n+1} = k a_{n} + \sqrt{\left(k^{2} - 1\right) a_{n}^{2} + 1}$ for $n = 0, 1, 2, \cdots$, with $k \geqslant 2$ where $k$ is a given positive integer. Find the general term of the sequence $\left\{a_{n}\right\}$.

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