
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
Bicycle travel is becoming increasingly popular, and various brands of mountain bikes are successively entering the market. The total sales of type $A$ cars operated by Shunfeng Carriage in June last year was $32,000. After being upgraded this year, the selling price of each type $A$ car increased by $400 compared to last year. If the number of type $A$ cars sold in June this year is the same as last year, the total sales of type $A$ cars in June this year will increase by $25\%$ compared to last year. The purchase and sale price table of type $A$ and type $B$ cars is as follows:

|                  | Type $A$ Car | Type $B$ Car |
|------------------|--------------|--------------|
| Purchase Price (in yuan per car) | $1100$ | $1400$ |
| Selling Price (in yuan per car) | This year's selling price | $2400$ |

$(1)$ Find the selling price per car of type $A$ cars in June this year;  
$(2)$ The car dealership plans to purchase a new batch of $50$ cars in July, including type $A$ and type $B$ cars, and the quantity of type $B$ cars purchased does not exceed twice the quantity of type $A$ cars. How should they purchase the cars to maximize profit?

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