
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 each term is non-zero, and the sum of the first $n$ terms is represented by $S\_n$. For any $n∈N^{*}$, the vector $\overrightarrow{m\_n}=(a_{n+1}-a_n, 2a_{n+1})$ is the normal vector of the line $y=kx$. If $\lim_{n\to \infty }S\_n$ exists, find the range of the real number $k$.

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