
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 three mascots of the Hangzhou Asian Games, "Cong Cong", "Chen Chen", and "Lian Lian", are named "Jiangnan Yi", which comes from the famous line of the Tang Dynasty poet Bai Juyi, "Jiangnan Yi, most memorable is Hangzhou". It integrates the historical and cultural heritage, natural ecology, and innovative genes of Hangzhou. The mascots were well-received as soon as they were launched. A certain store purchased a certain type of Asian Games mascot at a price of $35$ yuan each and sold them at a price of $58$ yuan each. According to statistics, the sales volume in April was $256$ units, and in June it was $400$ units.  
$(1)$ Calculate the monthly average growth rate of the sales volume of this mascot from April to June.  
$(2)$ According to market forecasts, the sales volume in July will be the same as in June. Now, in order to reduce inventory, the store is using a discount promotion method. A survey found that for this mascot, for every $1$ yuan reduction in price, the monthly sales volume would increase by $20$ units. At what price should the mascot be sold for the monthly sales profit to reach $8400$ yuan?

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