
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, and the even terms form a geometric sequence, both with a common difference and a common ratio of 2, respectively. Additionally, it is given that $a_2 + a_4 = a_1 + a_5$, and $a_7 + a_9 = a_8$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Find all positive integer values of $m$ for which $a_m \cdot a_{m+1} \cdot a_{m+2} = a_m + a_{m+1} + a_{m+2}$ holds true.

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