
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 a monotonic geometric sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$. If $S_3=39$, and $3a_4$ is the arithmetic mean of $a_6$ and $-a_5$.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;

(Ⅱ) If the sequence $\{b_n\}$ satisfies $b_n=\log_3{a_{2n+1}}$, and the sum of the first $n$ terms of $\{b_n\}$ is $T_n$, find the value of $\frac{1}{T_1}+\frac{1}{T_2}+\frac{1}{T_3}+\cdots+\frac{1}{T_n}$.

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