
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 sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, and $a_n = 2 - 2S_n$, and a sequence $\{b_n\}$ that forms an arithmetic sequence with $b_5 = 14$ and $b_7 = 20$:

$(1)$ Find the general term formula for the sequence $\{a_n\}$;
$(2)$ If $c_n = a_n \cdot b_n$ where $n \in \mathbb{N^*}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted $T_n$.

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