
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 a popular means of transportation for high school students, with a huge market and fierce competition. The total sales of type $A$ bikes operated by a certain brand dealer last year was $50,000. This year, the selling price per bike is $400 less than last year. If the quantity sold remains the same, the total sales will decrease by $20\%$.

Given the table below for the purchase and selling prices of type $A$ and type $B$ bikes this year:

|          | Type $A$       | Type $B$       |
|----------|---------------|---------------|
| Purchase Price | $1100$ yuan per bike | $1400$ yuan per bike |
| Selling Price  | $x$ yuan per bike   | $2000$ yuan per bike |

$(1)$ Let the selling price per type $A$ bike this year be $x$ yuan. Find the value of $x$.

$(2)$ The brand dealer plans to purchase a new batch of $A$ bikes and new type $B$ bikes, totaling $60$ bikes. The quantity of type $B$ bikes purchased should not exceed twice the quantity of type $A$ bikes. How should the quantities of the two types of bikes be arranged to maximize profit after selling this batch of bikes?

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