
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
Mountain bikes are becoming increasingly popular among high school students, with various brands entering the market. The total sales revenue of type $A$ bikes sold by a certain bike shop was $50,000$ yuan last year. This year, the selling price per bike is $400$ yuan lower than last year. If the quantity sold remains the same, the total sales revenue will decrease by $20\%$ compared to last year.
$(1)$ What is the selling price per bike of type $A$ this year? (Solve using the method of setting up equations)
$(2)$ The bike shop plans to bring in a new batch of $A$ type bikes and new model $B$ type bikes totaling $60$ bikes. The quantity of $B$ type bikes purchased should not exceed twice the quantity of $A$ type bikes. How should they purchase the bikes to maximize profit?
The purchase and selling prices of type $A$ and type $B$ bikes are shown in the table below:

|                  | Type $A$ | Type $B$ |
|------------------|----------|----------|
| Purchase Price (yuan) | $1100$   | $1400$   |
| Selling Price (yuan)  | This year's selling price | $2000$   |

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