
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
Compute the smallest value  $C$  such that the inequality   $$ x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C $$   holds for all real  $x$  and  $y$ .

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