
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
The sequence $\{a_n\}$ has a sum of the first $n$ terms denoted by $S_n$. Given $S_1=1, S_2=2$, and $S_{n+1}-3S_n+2S_{n-1}=0$ $(n\in\mathbb{N}^*)$ and $n\geqslant 2$, find $a_n$.

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