
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
Let \( x, y, z \) be the side lengths of a right-angled triangle, with \( z \) being the hypotenuse. Determine the maximum value of the parameter \( k \) for which the inequality \( x^{2}(y+z) + y^{2}(z+x) + z^{2}(x+y) > k \cdot x \cdot y \cdot z \) always holds. Also, indicate when equality occurs.

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