
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 "Jiangnan Yi" of the 2023 Hangzhou Asian Games integrates the historical and cultural heritage, natural ecology, and innovative genes of Hangzhou. The three mascots are named "Cong Cong," "Lian Lian," and "Chen Chen," with vivid and beloved designs. According to statistics, the sales volume on a certain e-commerce platform was 50,000 units in July and 72,000 units in September.  
$(1)$ If the monthly average growth rate from July to September on the platform is the same, what is the monthly average growth rate?  
$(2)$ Market research found that the cost price of a mascot doll in a certain store is $60 each. If the selling price is $100 each, and 20 units are sold per day, with an additional 20 units sold for every $10 decrease in price. In order to promote and advertise, the profit of each mascot doll must not exceed 30% of the cost price. Let the total profit from selling mascot dolls per day be $w (in dollars)$. What should be the selling price of each mascot doll for the store to maximize profit? What is the maximum profit in dollars?

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