
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 ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$. The sequence satisfies the conditions $a_1 = 1$ and $a_{n+1} = 2S_n + 1$. If the inequality $(S_n + \frac{1}{2}) \cdot k \geq \frac{1}{3}$ holds true for any $n \in \mathbb{N}^*$, determine the range of values for the real number $k$.

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