
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 $f(x)=(\log_2x)^2-\log_2x^2+3$, when $x\in[1,4]$, the maximum value of $f(x)$ is $m$, and the minimum value is $n$.

$(1)$ If the terminal side of angle $\alpha$ passes through point $p(m,n)$, find the value of $\sin \alpha+\cos \alpha$;

$(2)$ Let $g(x)=m\cos (nx+ \frac{\pi}{m})-n$, $h(x)=g(x)-k$ has two different zeros $x_1,x_2$ in $[0, \frac{\pi}{2}]$, find the range of $k$.

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