
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 all strictly positive real numbers \( c \) such that for all positive real numbers \( x, y, z \) the following inequality holds:

\[ \frac{x^4}{(y^2+1)(z^2+1)} + \frac{y^4}{(x^2+1)(z^2+1)} + \frac{z^4}{(x^2+1)(y^2+1)} + \frac{6}{1+c(x \sqrt{x}+y \sqrt{y}+z \sqrt{z})^{4/3}} > 3 \]

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