
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
Given a sequence \(a_{1}, a_{2}, a_{3}, \cdots\) such that \(a_{1} = 1\) and \(a_{n+1} = \frac{1}{16}\left(1 + 4a_{n} + \sqrt{1 + 24a_{n}}\right)\) for \(n = 1, 2, \cdots\), find a general formula for the \(n\)-th term of the sequence \(a_n\).

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