
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

The sum of the positive numbers \(a, b, c,\) and \(d\) is 4. Find the minimum value of the expression

$$
\frac{a^{8}}{\left(a^{2}+b\right)\left(a^{2}+c\right)\left(a^{2}+d\right)}+\frac{b^{8}}{\left(b^{2}+c\right)\left(b^{2}+d\right)\left(b^{2}+a\right)}+\frac{c^{8}}{\left(c^{2}+d\right)\left(c^{2}+a\right)\left(c^{2}+b\right)}+\frac{d^{8}}{\left(d^{2}+a\right)\left(d^{2}+b\right)\left(d^{2}+c\right)}
$$

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