
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
The agricultural company purchases a batch of agricultural products for sale at a price of $30$ yuan per kilogram. In order to determine the relationship between the daily sales volume $n$ (in kilograms) and the selling price $x$ (in yuan per kilogram), some data obtained from market research are shown in the table below:

| Selling Price $x$ (yuan per kilogram) | $30$ | $35$ | $40$ | $45$ | $50$ |
|---------------------------------------|------|------|------|------|------|
| Daily Sales Volume $n$ (kilograms)    | $600$| $450$| $300$| $150$| $0$  |

$(1)$ Based on the data in the table, determine the functional expression between $n$ and $x$ using the knowledge of linear functions, quadratic functions, and inverse proportionality functions. Write down the functional expression between $n$ and $x as ______.

$(2)$ How should the agricultural company determine the selling price of these agricultural products to maximize daily sales profit?

$(3)$ The agricultural company incurs a cost of $a$ yuan (where $a > 0$) for selling $1$ kilogram of this agricultural product. When $40 \leq x \leq 45$, the maximum value of the daily profit of the agricultural company is $2430$ yuan. Find the value of $a$. (Daily profit = Daily sales profit - Daily expenditure)

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