
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]


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A certain supermarket purchased two types of goods, A and B, for $6000$ yuan for the first time. The number of items of type B is $\frac{1}{2}$ more than the number of items of type A plus $15$. The cost and selling price of goods A and B are shown in the table below: (Note: Profit = Selling Price - Cost)

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

$(1)$ After selling all the goods A and B purchased for the first time, how much profit will the supermarket make in total?  
$(2)$ For the second time, the supermarket purchased goods A and B at the cost of the first time. The number of items of type A remains the same, while the number of items of type B is three times that of the first time. Goods A are sold at the original price, and goods B are sold at a discounted price. After selling all the goods for the second time, the total profit is $180$ yuan more than that of the first time. Find the discount rate at which goods B are sold for the second time.

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