
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
Let $a, b, c$ be positive real numbers. Find the minimum value of
\[
\frac{6a}{b + 2c} + \frac{6b}{c + 2a} + \frac{2c}{a + 2b} + \frac{6c}{2a + b}.
\]

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