
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
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, and it satisfies $S_n=2-a_n$ for $n=1,2,3,\ldots$

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;  
(Ⅱ) If the sequence $\{b_n\}$ satisfies $b_1=1$ and $b_{n+1}=b_n+a_n$, find the general formula for the sequence $\{b_n\}$;  
(Ⅲ) Given $c_n= \frac{n(3-b_n)}{2}$, find the sum of the first $n$ terms $T_n$ of $c_n$.

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