
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
For a sequence $\{a_n\}$ where each term is non-zero and the sum of the first $n$ terms is $S_n$. For any $n \in \mathbb{N}^*$, the vector $\overrightarrow{m_n} = (a_{n+1} - a_n, 2a_{n+1})$ is a normal vector to the line $y = kx$. If $\lim_{n \to \infty} S_n$ exists, then the range of the real number $k$ is \_\_\_\_\_\_.

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