
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
In order to hold a sports meeting, the junior high school of School A has purchased two types of sports equipment, $A$ and $B$. It is known that the unit price of type $A$ equipment is $10$ yuan less than that of type $B$ equipment. The school purchased an equal number of $A$ equipment for $300$ yuan and $B$ equipment for $360$ yuan. Based on the above information, answer the following questions:<br/>$(1)$ What are the unit prices of type $A$ and type $B$ equipment, respectively?<br/>$(2)$ With a budget of $1000$ yuan to purchase both type $A$ and type $B$ equipment, and exactly spending $1000$ yuan, please list all possible scenarios.

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