
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 that for any two real numbers $m$ and $n$, the function $max\{m,n\}=\left\{\begin{array}{l}m,&m≥n\\ n,&m＜n\end{array}\right.$ is defined. Let $f\left(x\right)=ax-2$ and $g\left(x\right)=x^{2}+bx-5$. <br/>$(1)$ If $a=2$ and $b=4$, let $h\left(x\right)=\max \{f\left(x\right),g\left(x\right)\}$. Find the minimum value of $h\left(x\right)$. <br/>$(2)$ If $a \gt 0$ and $b\in \mathbb{R}$, and $f\left(x\right)g\left(x\right)\geqslant 0$ always holds when $x \gt 0$, find the minimum value of $b+\frac{4}{a}$.

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