
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
In June, it is the season of abundant harvest of bayberries. A certain distributor sells bayberries at a cost of $8$ yuan per kilogram. When the selling price is set at $20$ yuan per kilogram, the distributor can sell $100$ kilograms per day. The distributor wants to increase profits by increasing sales volume. After a trial sale, it was found that for this type of bayberries, for every $1$ yuan decrease in selling price per kilogram, the daily sales volume increases by $20$ kilograms.  
$(1)$ When the selling price of bayberries decreases by how many yuan per kilogram, can the daily sales revenue reach $3000$ yuan?  
$(2)$ At what price per kilogram should the bayberries be set to maximize the distributor's daily profit? What is the maximum profit per day?

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