
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 certain store is having a promotion on two types of chocolates, $A$ and $B$. According to statistics, there were a total of 480 customers who came to buy these two types of chocolates on a certain day, with each customer buying one box of chocolate. The selling prices of types $A$ and $B$ chocolates are $90$ yuan and $50$ yuan, respectively.  
$(1)$ If the number of people choosing to buy type $B$ chocolate does not exceed $0.6$ times the number of people choosing to buy type $A$ chocolate, find the minimum number of people who choose to buy type $A$ chocolate.  
$(2)$ March 8th is International Women's Day, and the store expects more people to buy chocolates on that day. Therefore, they increased the selling price of type $A$ chocolate. It was found that the number of customers buying chocolates on Women's Day decreased compared to the minimum number of people choosing type $A$ chocolate as calculated in part $(1)$. For every $3$ yuan increase in the price of type $A$ chocolate, the number of people buying type $A$ chocolate decreased by 5, and the number of people buying type $B$ chocolate also decreased by 3. However, the selling price of type $B$ chocolate remained the same. Ultimately, the total sales of the two types of chocolates on Women's Day were the same as the total sales when the minimum number of people chose type $A$ chocolate as calculated in part $(1)$. Find the selling price of type $A$ chocolate on Women's Day.

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