
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 hot pot restaurant launched two double meal sets to attract customers. The table below shows the income statistics of the two sets in the past two days:

|          | Quantity of Set A | Quantity of Set B | Income |
|----------|-------------------|-------------------|--------|
| Day 1    | 20 times          | 10 times          | $2800$ |
| Day 2    | 15 times          | 20 times          | $3350$ |

$(1)$ Find the unit prices of these two sets.  
$(2)$ The cost of Set A is about $45$ yuan, and the cost of Set B is about $50 yuan. Due to material and seating limitations, the restaurant can supply a maximum of $50$ sets per day, and the quantity of Set A is not less than $\frac{1}{5}$ of the quantity of Set B. Find the maximum profit the restaurant can make each day from these two sets.  
$(3)$ The restaurant introduces value-added services where an additional $10$ yuan can be paid for each set to add extra ingredients. Customers can choose two out of four options: fish tofu, gluten, Sichuan noodles, and mushrooms. Xiaoming is a regular customer of this hot pot restaurant. In 2022, he spent a total of $1610$ yuan on purchasing two sets. If the quantity of Set A without extra ingredients accounts for $\frac{1}{4}$ of the total quantity, find the number of Set B with extra ingredients that Xiaoming chose.

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