
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
The 24th Winter Olympics was held in Beijing on February 4, 2022. A distributor predicted that two types of souvenirs with the mascot "Bing Dwen Dwen" might sell well. Therefore, the distributor purchased a total of 100 items of type A and type B souvenirs for a one-time cost of 6200 yuan. The cost and selling price of the two types of souvenirs are shown in the table below:

| Type | Cost (yuan per item) | Selling Price (yuan per item) |
|------|-----------------------|-------------------------------|
| A    | 50                    | 100                           |
| B    | 70                    | 90                            |

$(1)$ How many items of type A and type B souvenirs did the distributor purchase at once?  
$(2)$ If all the souvenirs are sold out on the day of the opening ceremony of the Beijing Winter Olympics, how much profit will be made?

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