
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 the function
\[ f(x, y, z) = \sqrt[5]{x+1} + \sqrt[5]{y+1} + \sqrt[5]{z+1}, \]
find the largest real number \(\lambda\) such that for any positive numbers \(x, y, z\) that satisfy \(x + y + z = 4\), we have \(f(x, y, z) > \lambda\).

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