
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 $f(x)=|mx+1|+|2x-1|$, where $m\in R$.  
$(Ⅰ)$ When $m=3$, find the solution set of the inequality $f(x) \gt 4$;  
$(Ⅱ)$ If $0 \lt m \lt 2$ and for any $x\in R$, $f(x)\geqslant \frac{3}{{2m}}$ always holds, find the minimum value of $m$.

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