
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 solution set of the inequality $|2x-1| < 1$ be $M$, and $a \in M, b \in M$.

$(1)$ Compare the size of $ab+1$ and $a+b$;

$(2)$ Let $\max A$ represent the maximum number in the set $A$, and $h=\max \left\{\frac{2}{\sqrt{a}},\frac{a+b}{\sqrt{ab}},\frac{2}{\sqrt{b}}\right\}$, find the range of $h$.

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