
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 maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.

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