
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 that the sequence $\{a_n\}$ is an arithmetic sequence, $a_1=t^2-t$, $a_2=4$, $a_3=t^2+t$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;  
$(2)$ If the sequence $\{a_n\}$ is increasing and the sequence $\{b_n\}$ satisfies $\log_2 b_n=a_n$, find the sum of the first $n$ terms $S_n$ of the sequence $\{(a_n-1)b_n\}$.

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