
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 sports equipment store purchaser needs to purchase a total of 100 basketballs and volleyballs from the manufacturer, with a total payment not exceeding $11,815. The wholesale prices from the two types of balls and the retail prices at the store are shown in the table below. Solve the following questions:

| Product | Wholesale Price (in $/each) | Retail Price (in $/each) |
|---------|-----------------------------|--------------------------|
| Basketball | $130 | $160 |
| Volleyball | $100 | $120 |

$(1)$ How many basketballs can the purchaser purchase at most?  
$(2)$ If the store sells all 100 balls at the retail price to ensure a profit of at least $2,580, how many basketballs should the purchaser buy at least, and how much profit can the store make at most?

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