
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]


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A certain middle school, in order to implement the requirements of the "Notice on Further Strengthening the Physical Fitness Management of Primary and Middle School Students" issued by the Ministry of Education, has decided to add basketball and soccer as elective courses, and needs to purchase a batch of basketballs and soccer balls. It is known that purchasing $2$ basketballs and $3$ soccer balls costs a total of $510$ yuan; purchasing $3$ basketballs and $5$ soccer balls costs a total of $810$ yuan.  
$(1)$ Find the unit prices of basketballs and soccer balls in yuan.  
$(2)$ The school plans to purchase a total of $50$ basketballs and soccer balls, with at least $30$ basketballs and a total cost not exceeding $5500$ yuan. What are the possible purchasing schemes?

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