
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 community supermarket purchased two types of goods, A and B, for $6000. The number of items of type A is 30 less than twice the number of items of type B. The cost and selling price of goods A and B are as shown in the table below:

|          | A   | B   |
|----------|-----|-----|
| Cost     | $22 | $30 |
| Selling Price | $29 | $40 |

$(1)$ How many items of goods A and B did the supermarket purchase?
$(2)$ How much profit can the supermarket make after selling all the goods A and B purchased for the first time?
$(3)$ The supermarket purchased goods A and B for the second time at the same cost as the first time. The number of items of type A remained the same, while the number of items of type B was three times that of the first time. Goods A were sold at the original price, but some of goods B were not selling well. Therefore, the supermarket decided to offer a 50% discount on the remaining goods B. If the supermarket made a total profit of $2350 during this sale, how many items of goods B were sold at a 50% discount?

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