
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 real number \( M \) for which the inequality \[ \mid ab(a^{2}-b^{2}) + bc(b^{2}-c^{2}) + ca(c^{2}-a^{2}) \mid \leq M (a^{2} + b^{2} + c^{2})^{2} \] holds for any real numbers \( a, b, \) and \( c \).

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