
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 two sequences $\{a_n\}$ and $\{b_n\}$, where $a_1= \frac{1}{2}$, the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=n^2a_n(n\geqslant 1)$, and the sequence $\{b_n\}$ satisfies $b_1=2$, $b_{n+1}=2b_n$.  
(Ⅰ) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.  
(Ⅱ) Does there exist a natural number $m$ such that for any $n\in\mathbb{N}^*$, $n\geqslant 2$, it always holds that $1+ \frac{1}{b_1}+ \frac{1}{b_2}+\ldots+ \frac{1}{b_{n-1}} < \frac{m-8}{4}$? If it exists, find the minimum value of $m$.  
(Ⅲ) If the sequence $\{c_n\}$ satisfies $c_n= \begin{cases} \frac{1}{na_n}, & \text{when } n \text{ is odd} \\ b_n, & \text{when } n \text{ is even} \end{cases}$, when $n$ is even, find the sum of the first $n$ terms of the sequence $\{c_n\}$, $T_n$.

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