
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
Recently, many cities in China have been intensifying efforts to develop the "night economy" to meet the diverse consumption needs of different groups and to boost employment, drive entrepreneurship, and enhance regional economic development vitality. A handicraft seller at a night market found through a survey of daily sales over the past month (30 days) that the selling price $P(x)$ per item (unit: yuan per item) approximately follows the function with respect to the $x$-th day $(1\leqslant x\leqslant 30, x\in \mathbb{N})$ as $P(x)=5+\frac{k}{x}$ (where $k$ is a positive constant). The daily sales volume $Q(x)$ per day (unit: items) for some days is shown in the table below:

| $x$   | $10$ | $15$ | $20$ | $25$ | $30$ |
|-------|------|------|------|------|------|
| $Q(x)$| $90$ | $95$ | $100$| $95$ | $90$ |

It is known that the daily sales revenue on the 10th day is $459$ yuan.

$(1)$ Find the value of $k$;  
$(2)$ Given the following four function models:  
① $Q(x)=ax+b$;  
② $Q(x)=a|x-m|+b$;  
③ $Q(x)=a^{x}+b$;  
④ $Q(x)=b\cdot \log ax$.  
Based on the data in the table, choose the most suitable function model to describe the relationship between the daily sales volume $Q(x)$ and the day $x$, and find the analytical expression of that function;  
$(3)$ Let the daily sales revenue of the handicraft item be a function $y=f(x)$ (unit: yuan). Find the minimum value of this function.

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