
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
For any triplet of non-negative numbers \( a, b, c \) such that \( a^{2}, b^{2}, c^{2} \) are the sides of a triangle, show the inequality:

$$
(a+b+c)\left(a^{2}+b^{2}+c^{2}\right)\left(a^{3}+b^{3}+c^{3}\right) \geqslant 4\left(a^{6}+b^{6}+c^{6}\right)
$$

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