
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 $\{a_{n}\}$ where all terms are positive, $a_{1}=1$, $a_{n+1} \gt a_{n}$, and $a_{n+2}+a_{n}=a_{2}\cdot a_{n+1}$ for $n\in N^{*}$.  
$(1)$ If the sequence $\{a_{n+1}-a_{n}\}$ forms an arithmetic progression, find the sum $S_{n}$ of the first $n$ terms of the sequence $\{a_{n}\}$.  
$(2)$ If the sequence $\{a_{n+1}-2a_{n}\}$ forms a geometric progression, and the sequence $\{a_{n}\}$ is not a geometric progression, find the general term of the sequence $\{a_{n}\}$.

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