
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 bakery has launched a new bread. The cost price of each bread is $4, and the selling price is $10. Only one batch of this bread is made each day (with a minimum of 15 and a maximum of 30 loaves). If the bread is not sold out by the end of the day, the remaining loaves are sold at $2 each. To determine the number of loaves in a batch, the bakery has recorded the daily demand for this new bread over the past 30 days and compiled the following table:

| Daily Demand | 15  | 18  | 21  | 24  | 27  |
|--------------|-----|-----|-----|-----|-----|
| Frequency    | 10  | 8   | 7   | 3   | 2   |

$(1)$ Replace the probability of each daily demand with the frequency of the daily demand recorded over 30 days. Calculate the probability that the number of loaves in a batch is at least 21.  
$(2)$ For the 30 days, the bakery bakes 21 loaves of this new bread each day.  
① If the daily demand is 15 loaves, calculate the daily profit for this new bread.  
② Calculate the average daily profit for this new bread over the 30 days.

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