
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 mascot of the 2023 Hangzhou Asian Games is a group of robots that carry profound cultural heritage and are full of contemporary vitality. They are composed of three robots named Congcong, Lianlian, and Chenchen, forming the "Jiangnan Yi" combination. "Jiangnan Yi" comes from the famous line of the Tang Dynasty poet Bai Juyi, "Remembering Jiangnan, the most unforgettable is Hangzhou," integrating Hangzhou's historical culture, natural ecology, and innovative genes. The three mascot designs are vivid and loved by everyone. A certain store purchases keychains with the mascot design at a price of $38$ yuan each and sells them at a price of $66$ yuan each. According to statistics, the sales volume in July was $256$ units, and in September it was $400$ units.
$(1)$ Calculate the monthly average growth rate of the sales volume of this keychain from July to September.
$(2)$ Starting from October, the store decides to use price reductions to promote sales and reward customers. Based on sales experience, it is predicted that for every $1$ yuan reduction in price, the monthly sales volume will increase by $20$ units. At what price should the keychain be sold for the monthly sales profit to be maximized, and what is the maximum profit?

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