
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
To further promote the theme of "green waters and lush mountains are mountains of gold and silver," a certain tourist scenic area, themed around "green waters and lush mountains," has specially produced a commemorative badge and decided to release it to the market soon. According to market research, the expected market price $y$ (in units of yuan) of each commemorative badge and the time of listing $x$ (in days) are as follows:

| Time of Listing $x$ (days) | $2$ | $6$ | $32$ |
|---------------------------|----|----|-----|
| Market Price $y$ (yuan)   | $148$ | $60$ | $73$ |

$(1)$ Based on the data in the table, select an appropriate function from $y=ax+b (a\neq 0)$, $y=\frac{a}{x}+b (a\neq 0)$, $y=a\log_{b}x (a\neq 0, b>0, b\neq 1)$, $y=ax+\frac{b}{x} (a>0, b>0)$ to describe the relationship between the market price $y$ of each commemorative badge and the time of listing $x$ (no need to provide a reason). Utilize the selected function to determine the number of days the badge should be listed to achieve the lowest market price and the corresponding lowest market price.

$(2)$ Let the function you selected be $y=f(x)$. If for any $x\in [k,+\infty) (k>0)$, the inequality $kf(x)-32k-210\geq 0$ always holds, find the range of real numbers for $k$.

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