
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 maximum real number \( k \) such that for any positive real numbers \( a, b, c \), the following inequality holds:
$$
\frac{(b-c)^{2}(b+c)}{a}+\frac{(c-a)^{2}(c+a)}{b}+\frac{(a-b)^{2}(a+b)}{c} 
\geqslant k\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)
$$

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