
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 $|x+1|+|2x-1|\leqslant 3$ with respect to $x$, the solution set is denoted as ${x|m\leqslant x\leqslant n}$.

(I) Find the values of real numbers $m$ and $n$;

(II) Suppose $a, b, c$ are all positive numbers, and $a+b+c=n-m$, find the minimum value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.

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