
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 such that \( a + b + c = 1 \). Find the minimum value of
\[
\frac{1}{a^2 + 2b^2} + \frac{1}{b^2 + 2c^2} + \frac{1}{c^2 + 2a^2}.
\]

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