
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
Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$

where " \( \sum \) " denotes a cyclic sum.

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