
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
Elective 4-5: Selected Topics on Inequalities

Given the function $f(x)=|x+1|+|x-1|$.

(Ⅰ) If there exists $x_0 \in \mathbb{R}$, such that the inequality $f(x_0) \leqslant m$ holds, find the minimum value $M$ of the real number $m$;

(Ⅱ) Under the condition of (1), if the positive numbers $a$, $b$ satisfy $3a+b=M$, find the minimum value of $\dfrac{1}{2a}+\dfrac{1}{a+b}$.

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