
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 real numbers \(a, b, c\) and non-negative real numbers \(x, y\) satisfy the conditions
\[ a^6 + b^6 + c^6 = 3 \]
and 
\[ (x + 1)^2 + y^2 \leq 2. \]

Find the minimum value of
\[ I = \frac{1}{2a^3 x + b^3 y^2} + \frac{1}{2b^3 x + c^3 y^2} + \frac{1}{2c^3 x + a^3 y^2}. \]

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