
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
It is known that the odd terms of the sequence $\{a_n\}$ form an arithmetic sequence with the first term being 1, and the even terms form a geometric sequence with the first term being 2. The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, and it satisfies $S_5 = 2a_4 + a_5$, $a_9 = a_3 + a_4$.
(1) Find the general formula for the sequence $\{a_n\}$;
(2) If $a_ma_{m+1} = a_{m+2}$, find the value of the positive integer $m$;
(3) Does there exist a positive integer $m$ such that $\frac{S_{2m}}{S_{2m-1}}$ is exactly one of the terms in the sequence $\{a_n\}$? If it exists, find all the values of $m$ that satisfy the condition; if not, explain why.

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