
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 unit is preparing to purchase two types of classified trash cans, $A$ and $B$, in response to the government's call. When making the purchase, it was found that the unit price of type $A$ is $50$ yuan less than the unit price of type $B$. It was observed that the number of type $A$ trash cans that can be purchased with $2000$ yuan is the same as the number of type $B$ trash cans that can be purchased with $2200$ yuan.
$(1)$ Find the unit prices of the two types of trash cans, $A$ and $B$.
$(2)$ If the unit needs to purchase $6$ classified trash cans and the total cost does not exceed $3100$ yuan, find all the different ways of purchasing.

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