
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 function $f(x)=|x+1|+a|2x-1|$.  
(Ⅰ) When $a= \frac{1}{2}$, if $f(x)\geqslant \frac{1}{m}+ \frac{1}{n}(m,n > 0)$ always holds for any $x∈\mathbb{R}$, find the minimum value of $m+n$;  
(Ⅱ) If the solution set of $f(x)\geqslant |x-2|$ contains $[-1,2]$, find the range of the real number $a$.

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