
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 sum of the first $n$ terms of the sequence $\{b_n\}$ be $S_n$, and $b_n = 2 - 2S_n$. The sequence $\{a_n\}$ is an arithmetic sequence, and $a_5 = 14$, $a_7 = 20$.  
(1) Find the general formula for the sequence $\{b_n\}$.  
(2) If $c_n = a_n \cdot b_n$ ($n=1,2,3,\ldots$), and $T_n$ is the sum of the first $n$ terms of the sequence $\{c_n\}$. Find $T_n$.

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