
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
A watermelon vendor purchases a batch of small watermelons at a price of $2$ yuan per kilogram and sells them at a price of $3$ yuan per kilogram. They can sell 200 kilograms per day. In order to promote sales, the vendor decides to reduce the price. After investigation, it was found that for every $0.1$ yuan reduction in price per kilogram, they can sell an additional 40 kilograms per day. In addition, the fixed costs such as rent amount to a total of $24$ yuan per day.  
$(1)$ If the price per kilogram of this type of watermelon is reduced by $x$ yuan, the daily sales volume will be ______ kilograms (expressed as an algebraic expression involving $x$);  
$(2)$ To make a profit of $200$ yuan per day from selling this fruit and to maximize the daily sales volume, how much should the price per kilogram be reduced by?

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