
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 maximum value of  $M$  for which for all positive real numbers  $a, b, c$  we have 
\[ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) \]

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