
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 smallest constant \( k \) so that for any positive real numbers \( x, y, \) and \( z \),
\[
\sqrt{\frac{x}{y + z + 2}} + \sqrt{\frac{y}{x + z + 2}} + \sqrt{\frac{z}{x + y + 2}} > k \cos\left(\frac{\pi}{6}\right).
\]

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