
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
Mr. Li, the owner of a fruit shop, purchased lychee and longan fruits for $3900$ yuan for the first time, and after selling them, he made a profit of $1200$ yuan. The cost price and selling price of the fruits are as shown in the table: (Total profit = profit per unit $\times$ quantity sold)

| Item Price | Lychee | Longan |
|------------|--------|--------|
| Cost Price (yuan/box) | $120$ | $100$ |
| Selling Price (yuan/box) | $150$ | $140$ |

$(1)$ How many boxes of lychee and longan fruits did Mr. Li purchase for the first time?  
$(2)$ For the second time, Mr. Li purchased lychee and longan fruits at the original price. The number of lychee boxes purchased remained the same, while the number of longan boxes purchased was twice that of the first time. Lychee was sold at the original price, while longan was sold at a discounted price. After selling both fruits, it was found that the profit earned was different from the first time, only $960$ yuan. What was the discount rate at which longan was sold?

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