
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
The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, with $a_1=1$ and $a_{n+1}=2S_n+1$. If for any $n\in\mathbb{N}^*$, $(S_n+ \frac{1}{2})\cdot k \geq \frac{1}{3}$ always holds, then the range of values for the real number $k$ is \_\_\_\_\_\_.

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