
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 chemical materials distribution company in Ezhou purchased a certain amount of chemical raw materials, priced at $30$ yuan per kilogram. The price department stipulates that the selling price should not exceed $60$ yuan per kilogram, nor be less than $30$ yuan per kilogram. Through market research, it was found that the daily sales volume $y$ (in kilograms) is a linear function of the selling price $x$ (in yuan), and when $x=60$, $y=80$; when $x=50$, $y=100$. In the sales process, an additional cost of $450$ yuan is incurred daily.<br/>$(1)$ Find the functional relationship between $y$ and $x$, and write down the range of values for the independent variable $x$.<br/>$(2)$ Find the functional relationship between the daily profit $w$ (in yuan) of the company selling the raw material and the selling price $x$ (in yuan).<br/>$(3)$ At what price per unit should the company sell to maximize its daily profit? What is the maximum profit?

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