
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 positive sequence $\{a_n\}$ satisfies $\frac{a_{n+1}}{a_{n-1}} + \frac{a_{n-1}}{a_{n+1}} = \frac{4a_n^2}{a_{n+1}a_{n-1}} -2$ $(n \geqslant 2, n \in \mathbb{N}^*)$, and $a_6=11$, the sum of the first $9$ terms is $81$.  
(I) Find the general formula for the sequence $\{a_n\}$.  
(II) If the sum of the first $n$ terms of the sequence $\{\lg b_n\}$ is $\lg (2n+1)$, let $c_n= \frac{a_n \cdot b_n}{2^{n+1}}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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