
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 supermarket deals with two types of products, $A$ and $B$. The cost of product $A$ per kilogram is $20$ yuan. Through trial sales, it was found that the daily sales quantity $y$ (in kilograms) of this product and the unit price $x$ (in yuan per kilogram) satisfy a linear function relationship. The corresponding values of daily unit price and sales quantity are shown in the table below:

| Unit Price $x$ (yuan per kilogram) | $25$ | $30$ | $35$ | $40$ |
|------------------------------------|------|------|------|------|
| Sales Quantity $y$ (kilograms)      | $50$ | $40$ | $30$ | $20$ |

The cost of product $B$ is $6$ yuan per kilogram, with a unit price of $10$ yuan per kilogram. However, the total daily supply of product $B$ is only $60$ kilograms, which can be sold out on the same day. In order to benefit consumers, the supermarket has launched a "buy one get one free" promotion, which means buying $1$ kilogram of product $A$ will get $1$ kilogram of product $B$ for free.

$(1)$ Find the function expression between $y$ (in kilograms) and $x$ (in yuan per kilogram);  
$(2)$ Let the total daily profit from the sales of these two products be $w$ yuan. Find the relationship between $w$ and $x$;  
$(3)$ If the selling price of product $A$ is not lower than the cost and not higher than $180\%$ of the cost, at what price should the unit price be set to maximize the total profit of the day? What is the maximum profit? (Total profit = total sales of the two products - cost of the two products)

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