
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 $|x+3|+|x+m|\geqslant 2m$ with respect to $x$, where the solution set is $\mathbb{R}$.
(1) Find the maximum value of $m$.
(2) Given $a > 0$, $b > 0$, $c > 0$, and $a+b+c=1$, find the minimum value of $2a^{2}+3b^{2}+4c^{2}$ and the corresponding values of $a$, $b$, and $c$.

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