
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 functions $f(x)=\log_{2}x$ and $g(x)=-\frac{1}{2}x+4$.  
$(1)$ Find the value of $g(f(\frac{1}{4}))$.  
$(2)$ Choose one of the following to answer.  
① For all $a, b \in \mathbb{R}$, define $max\{a,b\}=\left\{\begin{array}{l}a, a \geq b\\ b, a < b\end{array}\right.$, find the analytical expression of $h(x)=\max \{f(x),g(x)\}$ and write the minimum value of $h(x)$.  
② For all $a, b \in \mathbb{R}$, define $min\{a,b\}=\left\{\begin{array}{l}a, a \leq b\\ b, a > b\end{array}\right.$, find the analytical expression of $h(x)=\min \{f(x),g(x)\}$ and write the maximum value of $h(x)$.

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