
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
Determine the smallest possible real constant  $C$  such that the inequality  $$ |x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1| $$  holds for all real numbers  $x, y, z$  satisfying  $x + y + z = -1$ .

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