
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 smallest real number \(M\) such that for all real numbers \(a\), \(b\), and \(c\), the following inequality holds:
$$
\begin{aligned}
& \left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \\
\leqslant & M\left(a^{2}+b^{2}+c^{2}\right)^{2} .
\end{aligned}
$$

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