
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
Before the Mid-Autumn Festival, a supermarket purchased a gift box of mooncakes for $40$ yuan per box at a cost of $180$ boxes, and sold them all at a price of $52$ yuan per box. According to market research by sales staff, for every $1$ yuan decrease in price from the base price of $52$ yuan, the sales volume will increase by $10$ boxes. The supermarket plans to purchase the same gift box for the second time, but not exceeding $210$ boxes.  
$(1)$ With the purchase price unchanged, if the price is reduced by $a$ yuan in the second sale compared to the first sale, then the profit per box of mooncake in the second sale is ______ yuan, and the expected sales volume is ______ boxes.  
$(2)$ Under the conditions of $(1)$, if the profit from the second sale is $2000$ yuan, find the value of $a$.

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