
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 largest constant  $K>0$  such that for any  $0\le k\le K$  and non-negative reals  $a,b,c$  satisfying  $a^2+b^2+c^2+kabc=k+3$  we have  $a+b+c\le 3$ .

(Dan Schwarz)

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