
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]


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Let $\left\{ a_n \right\}$ be a positive geometric sequence with $a_4=81$, and the arithmetic mean of $a_2, a_3$ is $\frac{3}{2}(a_1+a_2)$.

$(1)$ Find the general formula for the sequence $\left\{ a_n \right\}$;

$(2)$ Let $b_n=\log_3 a_{2n-1}$, and let $S_n$ be the sum of the first $n$ terms of the sequence $\left\{ b_n \right\}$. The sequence $\left\{ c_n \right\}$ satisfies $c_n=\frac{1}{4S_n-1}$. Find the sum of the first $n$ terms of the sequence $\left\{ c_n \right\}$, denoted as $T_n$.

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