
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
Xiao Ming made three purchases of products $A$ and $B$ at a certain store. Only one of the purchases had a discount on both products, while the other two were bought at regular prices. The quantities and costs of purchasing products $A$ and $B$ three times are shown in the table below:

|              | Quantity of Product $A$ | Quantity of Product $B$ | Price of Products $A$ and $B$ (yuan) |
|--------------|-------------------------|-------------------------|--------------------------------------|
| First Purchase  | 6                       | 5                       | 1140                                 |
| Second Purchase | 3                       | 7                       | 1110                                 |
| Third Purchase  | 9                       | 8                       | 1062                                 |

$(1)$ Find the original prices of products $A$ and $B$;  
$(2)$ If the discounts on products $A$ and $B$ are the same, how much discount is the store offering on these two products?

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