
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
Eating zongzi during the Dragon Boat Festival is an ancient traditional custom in China. A certain zongzi wholesale store sells each zongzi with a profit of $2$ yuan. According to the staff situation, they can make a maximum of $1100$ zongzi per day. Based on market research, if the price of each zongzi is reduced by $x$ yuan, the function relationship of the daily sales volume $y$ (units) of zongzi with respect to $x$ (yuan) is $y=800x+400$.  
$(1)$ If the price of each zongzi is reduced by $0.2$ yuan, then the store's daily sales volume will be ______ units, and the total profit per day will be ______ yuan.  
$(2)$ When the price of each zongzi is reduced by how many yuan, the store's daily total profit is exactly $1200$ yuan?

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