
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
If $\{a_n\}$ is a positive increasing geometric sequence, and $T_n$ represents the product of its first $n$ terms, and $T_4 = T_8$, then when $T_n$ takes the minimum value, the value of $n$ is ______.

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