
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 shopping mall plans to purchase a batch of basketballs and soccer balls. The unit price of basketballs is $30$ yuan more than the unit price of soccer balls. It is known that the quantity of soccer balls purchased for $360$ yuan is equal to the quantity of basketballs purchased for $480$ yuan. The selling price of basketballs is $150$ yuan each, and the selling price of soccer balls is $110$ yuan each.  
$(1)$ What are the unit prices of basketballs and soccer balls?  
$(2)$ The quantity of soccer balls sold by the mall is $10$ more than one-third of the quantity of basketballs sold, and the profit exceeds $1300$ yuan. How many basketballs must be sold at least?  
$(3)$ The mall plans to purchase a total of $100$ balls for no more than $10350$ yuan, with at least $43$ basketballs. How many purchasing schemes does the mall have? Which scheme yields the maximum profit?

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