
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
In the 2022 Beijing Winter Olympics, the Winter Paralympics have successfully concluded, and the lively and kind "Bing Dwen Dwen" and the auspicious and harmonious "Shuey Rongrong" have been loved by the general public. A supermarket purchased two mascots as souvenirs for the Winter Olympics and sold them. The table below shows the sales situation in the past two weeks:

| Sales Period | Sales Quantity of "Bing Dwen Dwen" | Sales Quantity of "Shuey Rongrong" | Sales Revenue |
|--------------|-----------------------------------|-----------------------------------|---------------|
| Week 1       | 3 units                           | 5 units                           | $1800         |
| Week 2       | 4 units                           | 10 units                          | $3100         |

$(1)$ Find the unit price of "Bing Dwen Dwen" and "Shuey Rongrong";  
$(2)$ If a company is prepared to purchase a total of 30 mascots of "Bing Dwen Dwen" and "Shuey Rongrong" with an amount not exceeding $6700, find the maximum number of "Bing Dwen Dwen" that can be purchased.

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