
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 all real numbers \(k\) such that the inequality
\[ 
a^{3} + b^{3} + c^{3} + d^{3} + 1 \geq k(a + b + c + d)
\]
holds for any \(a, b, c, d \in [-1, +\infty)\).

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