
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 $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, $a_1=2$, and $4S_n=a_n \cdot a_{n+1}$, for the sequence $\{b_n\}$, $b_1= \frac{1}{4}$, and $b_{n+1}= \frac{nb_{n}}{(n+1)-b_{n}}$, where $n\in\mathbb{N}^*$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $c_n= \frac{a_{n}}{2^{\frac{1}{3b_{n}}+ \frac{2}{3}}}$ ($n\in\mathbb{N}^*$), find the sum of the first $n$ terms of $\{c_n\}$, denoted as $T_n$.

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