
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 the geometric progression with positive terms $\{a_n\}$, the geometric mean of $a_3$ and $a_4$ is $\int_{\frac{1}{e}}^{e} \frac{1}{x}dx$. Find $T_6 = a_1 \cdot a_2 \cdot a_3 \cdot a_4 \cdot a_5 \cdot a_6$.

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