
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
Let $m$ positive numbers $a_1, a_2, …, a_m$ ($m \geq 4, m \in \mathbb{N}^*$) be arranged in a circle. Among them, $a_1, a_2, a_3, …, a_{k-1}, a_k$ ($k < m, k \in \mathbb{N}^*$) form an arithmetic sequence with a common difference $d$, and $a_1, a_m, a_{m-1}, …, a_{k+1}, a_k$ form a geometric sequence with a common ratio $q$.  
(1) If $a_1=d=1$, $q=2$, and $k=8$, find the sum of all terms $S_m$ of the sequence $a_1, a_2, …, a_m$;  
(2) If $a_1=d=q=3$ and $m < 2015$, find the maximum value of $m$;  
(3) When $q=2$, does there exist a positive integer $k$ such that $a_1+a_2+…+a_{k-1}+a_k = 3(a_{k+1}+a_{k+2}+…+a_{m-1}+a_m)$? If it exists, find the value of $k$; if it does not exist, explain the reason.

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