
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 \(a, b, c\) be the side lengths of a right triangle, with \(a \leqslant b < c\). Determine the maximum constant \(k\) such that the inequality \(a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c\) holds for all right triangles, and specify when equality occurs.

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