
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 an geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, and that $a_n$ is the arithmetic mean between $S_n$ and $2$, and an arithmetic sequence $\{b_n\}$ where $b_1=2$, the point $P(b_n, b_{n+1})$ lies on the graph of the linear function $y=x+2$. 
1. Find the general terms $a_n$ and $b_n$ for the sequences $\{a_n\}$ and $\{b_n\}$ respectively; 
2. Let $c_n = a_n \cdot b_n$, find the sum of the first $n$ terms, denoted as $T_n$, for the sequence $\{c_n\}$.

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