
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 the inequality about $x$, $|x-3|+|x-m| \geq 2m$, has the solution set $\mathbb{R}$.

(Ⅰ) Find the maximum value of $m$;  
(Ⅱ) Given $a>0$, $b>0$, $c>0$, and $a+b+c=m$, find the minimum value of $4a^2+9b^2+c^2$ and the values of $a$, $b$, $c$ at this point.

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