
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
Positive numbers \( a, b, c \) satisfy \( a^2 b + b^2 c + c^2 a = 3 \). Find the minimum value of the expression:

\[ A = \frac{\sqrt{a^6 + b^4 c^6}}{b} + \frac{\sqrt{b^6 + c^4 a^6}}{c} + \frac{\sqrt{c^6 + a^4 b^6}}{a}. \]

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