
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n = 2a_n - 2$ $(n = 1, 2, 3, \ldots)$, $(a_n \neq 0)$. In the sequence $\{b_n\}$, $b_1 = 1$, and point $P(b_n, b_{n+1})$ lies on the line $x - y + 2 = 0$.  
$(1)$ Find the general terms $a_n$ and $b_n$ for the sequences $\{a_n\}$ and $\{b_n\}$.  
$(2)$ Let $c_n = a_n \cdot b_n$, find the sum of the first $n$ terms $T_n$ of the sequence $\{c_n\}$.

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