
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
In the sequence $\{a_n\}$, it is known that $a_1=1$, $a_2=a$, and $a_{n+1}=k(a_n + a_{n+2})$ holds for any $n \in \mathbb{N}^*$, where the sum of the first $n$ terms of sequence $\{a_n\}$ is denoted as $S_n$. (Here $a$, $k$ are real numbers)  
(1) If sequence $\{a_n\}$ is an arithmetic sequence, find $S_n$;  
(2) If $a=1$ and $k=-\frac{1}{2}$, find $S_n$;  
(3) Does there exist a real number $k$ such that the sequence $\{a_n\}$ is a geometric sequence with a common ratio not equal to 1, and any three adjacent terms $a_m$, $a_{m+1}$, $a_{m+2}$, can be arranged in some order to form an arithmetic sequence? If it exists, find all the values of $k$; if not, explain why.

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