
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 printing factory produces two types of exercise books, A and B, totaling 400,000 copies per month, and all the exercise books are sold that month, with costs and selling prices as shown in the table below:

| Type | A   | B   |
|------|-----|-----|
| Cost | $1.2$ per copy | $0.4$ per copy |
| Price | $1.6$ per copy | $0.6$ per copy |

$(1)$ If the profit of the printing factory in May is $110,000, find out how many copies of type A and type B exercise books were produced respectively.

$(2)$ A school plans to spend $7680 to purchase exercise books from the printing factory. After negotiation, the company agrees to give a 10% discount on the selling price of type A exercise books, while no discount is given for type B exercise books. If the school can purchase 10,000 copies without overspending, what is the maximum number of type A exercise books that can be purchased?

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