
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 $|x+3|+|x+m| \geq 2m$ with respect to $x$ and its solution set is $\mathbb{R}$.

1. Find the range of values for the real number $m$.
2. Given $a > 0$, $b > 0$, $c > 0$ and $a + b + c = m$, find the minimum value of $a^2 + 2b^2 + 3c^2$ and the corresponding values of $a$, $b$, and $c$ when $m$ is at its maximum.

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