
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, entrepreneurship, and regional economic development. A handicraft seller at a night market conducted a survey on the daily sales of a particular handicraft over the past month (30 days). The relationship between the selling price $P(x)$ (in yuan) of the handicraft and time $x$ (in days) can be approximated by the function $P(x) = 10 + \frac{k}{x}$, where $k$ is a positive constant. The daily sales volume $Q(x)$ (in units) for some 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. Find the value of $k$.
2. Given the following four function models:
   - $Q(x) = ax + b$
   - $Q(x) = a|x-m| + b$
   - $Q(x) = a \cdot b^{x}$
   - $Q(x) = a \cdot \log_{b}x

   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 time $x$, and determine the analytical expression of that function.
3. 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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