
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 most well-preserved tradition during the traditional Mid-Autumn Festival is the custom of giving mooncakes as gifts. Mooncakes have become a symbol of the Mid-Autumn Festival and are also one of the most beloved pastries by the people of Foshan. Baghe Supermarket sells mooncakes of the $NF$ brand, with an average daily sales of 20 boxes, and a profit of $40$ yuan per box. As the Mid-Autumn Festival approaches, in order to increase sales and profits, Baghe has taken measures to reduce prices. After a period of time, it was found that for every $1$ yuan reduction in unit price, an average of 2 more boxes could be sold per day.  
$(1)$ If the price is reduced by $10$ yuan, the average daily sales quantity will be ______ boxes;  
$(2)$ To quickly reduce inventory, in order for Baghe Supermarket to have a daily sales profit of $1200$ yuan, how much should the price of each box of mooncakes be reduced by?

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