
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 $\{a_n\}$ satisfying $\frac {a_{n+1}}{a_{n-1}}+ \frac {a_{n-1}}{a_{n+1}}= \frac {4a_{n}^{2}}{a_{n+1}a_{n-1}}-2$ ($n\geqslant 2,n\in\mathbb{N}^{*}$), 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 $\lg (2n+1)$, and 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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