
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

Given that the real number \( m \) satisfies the following condition for the real coefficient quadratic equation \( a x^{2}+b x+c=0 \) with real roots,

$$(a-b)^{2}+(b-c)^{2}+(c-a)^{2} \geqslant m a^{2}$$

Then find the maximum value of \( m \).

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