
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
Find the largest constant \( c \) such that for all real numbers \( x \) and \( y \) satisfying \( x > 0, y > 0, x^2 + y^2 = 1 \), the inequality \( x^6 + y^6 \geq cx y \) always holds.

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