
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 \(\{a_{n}\}\) where the odd terms form an arithmetic sequence with the first term 1, and the even terms form a geometric sequence with the first term 2. The sum of the first \(n\) terms of the sequence is \(S_{n}\), and it satisfies \(S_{5}=2a_{4}+a_{5}\) and \(a_{9}=a_{3}+a_{4}\).

1. Find the general term formula of the sequence \(\{a_{n}\}\).
2. Given \(a_{m}a_{m+1}=a_{m+2}\), find the value of the positive integer \(m\).
3. Determine whether there exists a positive integer \(m\) such that \(\frac{S_{2m}}{S_{2m-1}}\) is exactly one term in the sequence \(\{a_{n}\}\). If it exists, find all values of \(m\) that satisfy the condition; if not, provide a reason.

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