
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
With the rapid development of the tourism industry, the demand for accommodation from foreign tourists has significantly increased, and the number of beds in a certain hotel continues to grow.  
$(1)$ The number of beds in the hotel increased from 200 at the end of 2021 to 288 at the end of 2023. Find the average annual growth rate of the number of beds in the hotel over these two years (from the end of 2021 to the end of 2023).  
$(2)$ The hotel plans to sell a souvenir craft to tourists, with a cost of $50 per item. In order to set a reasonable price, it is being test-marketed. According to market research, when the selling price is $100, the daily sales volume is 50 items. If the selling price is reduced by $1, an additional 5 items can be sold per day. If the hotel wants the daily sales profit to reach $4000 and maximize the sales volume, how should they set the price?

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