
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
Given that the sequence $\{a_n\}$ is a geometric sequence, and $S_n$ is the sum of its first $n$ terms. Let $T_n = S_1 + S_2 + \ldots + S_n$. If $a_2 \cdot a_3 = 2a_1$, and the arithmetic mean of $a_4$ and $2a_7$ is $\frac{5}{4}$, then $T_6 = \_\_\_\_\_\_$.

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