
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 positive real number  $k$  such that the inequality  $$ a^3+b^3+c^3-3\ge k(3-ab-bc-ca) $$  holds for all positive real triples  $(a;b;c)$  satisfying  $a+b+c=3.$ 

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