
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 sequences \(\{a_{n}\}\) and \(\{b_{n}\}\) satisfy:
\[ 
b_{n}=\begin{cases} a_{\frac{n+1}{2}}, & n \text{ is odd;} \\
\sqrt{a_{n+1}}, & n \text{ is even.}
\end{cases}
\]
If \(\{b_{n}\}\) is a geometric sequence and \(a_{2}+b_{2}=108\), find the general term formula for the sequence \(\{a_{n}\}\).

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