
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 denoted as $S_n$. Given $a_1= \frac{1}{2}$ and $2a_{n+1}+S_n=0$ for $n=1, 2, 3, \ldots$. If $S_n \leqslant k$ always holds, then the minimum value of $k$ is \_\_\_\_\_\_.

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