
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
In a geometric sequence $\{a_n\}$ where each term is a positive number, if $a_{m+1} \cdot a_{m-1} = 2a_m$ (with $m \geq 2$), and the product of the first $n$ terms of $\{a_n\}$ is denoted as $T_n$, given that $T_{2m-1} = 512$, determine the value of $m$.

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