
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 functions $f(x)=3-2\log_{2}x$ and $g(x)=\log_{2}x$  
$(1)$ If $x\in[1,2]$, find the range of the function $h(x)=[f(x)+1]g(x)$;  
$(2)$ Find the maximum value of the function $M(x)= \dfrac {f(x)+g(x)-|f(x)-g(x)|}{2}$;  
$(3)$ If for any $x\in[1,2]$, the inequality $f(x^{2})f( \sqrt {x}) > k\cdot g(x)$ always holds, find the range of the real number $k$.

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