
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
Let the functions be $f(x)=\frac{1}{x}$ $(x > 0)$, $g(x)=\log_{2}(2-|x+1|)$.

$(1)$ Write down the monotonic intervals of the function $g(x)$.

$(2)$ If $y=a$ has exactly one common point $M$ with the graph of the function $g(x)$, and $N$ is a moving point on the graph of $f(x)$. Find the minimum value of $|MN|$.

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