
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
Show the following inequality in two different ways:

$$
\forall a, b, c \in \mathbb{R}_{+}^{*}, \frac{a}{\sqrt{a^{2}+8 b c}} + \frac{b}{\sqrt{b^{2}+8 c a}} + \frac{c}{\sqrt{c^{2}+8 a b}} \geq 1
$$

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