
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 largest constant \( m \) such that for any positive real numbers \( x, y, z, \) and \( w \)
\[
\sqrt{\frac{x}{y+z+x}} + \sqrt{\frac{y}{x+z+w}} + \sqrt{\frac{z}{x+y+w}} + \sqrt{\frac{w}{x+y+z}} > m.
\]

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