
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
Selected Exercise $(4-5)$: Inequality Proof Lecture

Given the inequality $|x-3|+|x-m|\geqslant 2m$ with respect to $x$ and its solution set is $\mathbb{R}$.

(I) Find the maximum value of $m$;

(II) Given that $a > 0, b > 0, c > 0$, and $a + b + c = m$, find the minimum value of $4a^2 + 9b^2 + c^2$ and the corresponding values of $a, b, c$.

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