
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 its first $n$ terms sum $S\_n$, where $a\_1=1$, $a_{n+1}=λS\_n+1 (n∈N^*, λ≠ -1)$, and $a\_1$, $2a\_2$, $a\_3+3$ are the first three terms of an arithmetic sequence $\{b\_n\}$.

(1) Find the general terms of the sequences $\{a\_n\}$ and $\{b\_n\}$;

(2) Find the sum of the first $n$ terms, $T\_n$, of the sequence $\{a\_n b\_n\}$.

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