
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
A certain aquatic product dealer purchases a batch of a certain species of freshwater fish at a price of $30$ yuan per kilogram. Based on sales experience, the daily sales volume $y$ (in kilograms) of this species of freshwater fish is known to have a linear relationship with the selling price $x$ (in yuan per kilogram) $(30\leqslant x \lt 60)$. Some of the data is shown in the table below:

| Selling Price $x$ (yuan per kilogram) | $50$ | $40$ |
|---------------------------------------|------|------|
| Daily Sales Volume $y$ (kilograms)    | $100$| $200$|

$(1)$ Find the expression of $y$ in terms of $x$.  
$(2)$ Suppose the daily sales profit of this dealer for selling this species of freshwater fish is $W$ yuan. Neglecting other factors, determine at what selling price $x$ the daily sales profit $W$ is maximized. What is the maximum daily sales profit in yuan?

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