
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 sports goods store purchases two models of basketballs, $A$ and $B$, with the selling prices and purchase prices of these two models of basketballs as shown in the table below: If the store purchases 20 $A$ type basketballs and 30 $B$ type basketballs for $3400$ yuan; and purchases 30 $A$ type basketballs and 40 $B$ type basketballs for $4800$ yuan.<br/>$(1)$ Find the purchase prices of the two types of basketballs $A$ and $B$:<br/>$(2)$ In order to carry out campus basketball activities, a certain school plans to purchase $m$ $A$ type basketballs and $n$ $B$ type basketballs from the store at once, spending a total of $5400$ yuan. How much profit can the store make?<br/>$(3)$ In order to increase sales, the store implements a promotion: "Buy a basketball, get a jump rope for free." Package $1$: Buy $1$ $A$ type basketball and get $1$ jump rope for free; Package $2$: Buy $3$ $B$ type basketballs and get $1$ jump rope for free. It is known that the cost of each jump rope is $10$ yuan. On a certain day, the total profit from selling these two packages is $600$ yuan. How many $A$ and $B$ type basketballs were sold that day?

| Type | Purchase Price (yuan per unit) | Selling Price (yuan per unit) |
|------|--------------------------------|-------------------------------|
| $A$  | $x$                            | $120$                         |
| $B$  | $y$                            | $90$                          |

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