
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 greatest constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the following inequality holds:
\[ 
a^{2} b + b^{2} c + c^{2} d + d^{2} a + 4 \geqslant k \left(a^{2} + b^{2} + c^{2} + d^{2}\right).
\]

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