
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 $f(x)=|x+1|+|x-1|$.
(1) If there exists $x\_0∈R$ such that the inequality $f(x\_0)≤m$ holds, find the minimum value $M$ of the real number $m$;
(2) Under the condition of (1), if positive numbers $a$ and $b$ satisfy $3a+b=m$, find the minimum value of $\frac{1}{2a}+\frac{1}{a+b}$.

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