
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 $a\_1=4$, $a_{n+1}=S_n$, $n∈\mathbb{N}^*$.
(I) Find the values of $a_2$, $a_3$, $a_4$;
(II) Find the general term formula for the sequence $\{a_n\}$;
(III) Given an arithmetic sequence $\{b_n\}$ where $b_2=a_2$, $b_3=a_3$, find the sum of the first $n$ terms, $T_n$, for the sequence $\{a_n⋅b_n\}$.

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