
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 $\lambda$ such that the inequality $x y\left(x^{2}+y^{2}\right)+y z\left(y^{2}+z^{2}\right)+z x\left(z^{2}+x^{2}\right) \leqslant \lambda(x+y+z)^{4}$ holds for all non-negative real numbers $x, y, z$.

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