
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 county in our city organized 40 cars to transport a total of 200 tons of three types of apples, $A$, $B$, and $C, to be sold in other places. According to the plan, all 40 cars need to be loaded, each car can only carry one type of apple, and must be fully loaded. Based on the information provided in the table below, answer the following questions.

| Type of Apple | $A$ | $B$ | $C$ |
|---------------|-----|-----|-----|
| Tons per Car  | $6$ | $5$ | $4$ |
| Profit per Ton| $5$ | $7$ | $8$ |

$(1)$ Let the number of cars transporting type $A$ apples be $x$, and the number of cars transporting type $B$ apples be $y$. Find the functional relationship between $y$ and $x$.

$(2)$ If the number of cars transporting each type of apple is not less than 4, how many possible arrangements are there for the cars?

$(3)$ Under the condition in $(2)$, if we want to maximize the profit from this sale, which arrangement should be adopted? And calculate the maximum profit.

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