
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 $a+b=1$, for $\forall a$, $b\in(0,+\infty)$  
$(1)$ Find the minimum value of $\dfrac {1}{a}+ \dfrac {4}{b}\geqslant |2x-1|-|x+1|$ denoted as $M$.  
$(2)$ If $M\geqslant |2x-1|-|x+1|$ always holds, find the range of $x$.

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