
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 levels and to boost employment, drive entrepreneurship, and enhance regional economic development vitality. Night economy has become an important driving factor for urban economic development. According to the "2023 China Urban Night Economy Development Report" released by the Urban Research Institute, Fuzhou was selected as a second-tier city in the "TOP100 Cities with Prosperous Night Economy in China." A handicraft seller at Guangming Port Night Market found through a survey of daily sales over the past month (30 days) that the relationship between the selling price $P(x)$ per item (in yuan) and time $x$ (in days) approximately satisfies $P(x) = 10 + \frac{k}{x}$ (where $k$ is a positive constant). The partial data of the daily sales volume $Q(x)$ per day (in units) with respect to time $x$ (in days) is shown in the table below:

| $x$ | $10$ | $15$ | $20$ | $25$ | $30$ |
|-----|------|------|------|------|------|
| $Q(x)$ | $50$ | $55$ | $60$ | $55$ | $50$ |

It is known that the daily sales revenue on the 10th day is $505$ yuan.

$(1)$ Given the following three function models:
① $Q(x) = ax + b$;
② $Q(x) = a|x-m| + b$;
③ $Q(x) = a \cdot b^{x}$.
Please choose the most suitable function model from the data in the table to describe the relationship between the daily sales volume $Q(x)$ and time $x$, and find the analytical expression of that function.

$(2)$ Let the daily sales revenue of the handicraft be $f(x)$ (in yuan). Find the minimum value of $f(x)$.

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