
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 paint processing factory has 120 tons of material $A$ and 90 tons of material $B$. It is planned to produce two types of coatings, A and B, totaling 150 tons, using these two materials. It is known that producing one ton of type A coating requires 0.6 tons of material $A$ and 0.7 tons of material $B$, yielding a profit of 450 yuan; producing one ton of type B coating requires 0.9 tons of material $A$ and 0.4 tons of material $B$, yielding a profit of 500 yuan. If the production of type A coating is $x$ tons, the total profit $y$ obtained from producing these two coatings with this batch of materials is to be determined. <br/>$(1)$ Find the functional relationship between $y$ and $x$, and determine the range of values for the independent variable $x$; <br/>$(2)$ In the production of this batch of coatings at the paint processing factory, how many tons of type A coating should be produced to maximize profit? What is the maximum profit?

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