
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
Snail noodles are a traditional snack in Guangxi, with a unique flavor of spicy, refreshing, fresh, sour, and hot. Xiaoying bought some boxes of type $A$ snail noodles for the first time with $800$ yuan, and then bought some boxes of type $B$ snail noodles for the second time with $900$ yuan. It is known that the unit price of type $B$ snail noodles is $1.5$ times the unit price of type $A$ snail noodles, and the quantity of type $B$ snail noodles purchased for the second time is $2$ boxes less than the quantity of type $A$ snail noodles purchased for the first time.  
$(1)$ Find the unit price of each type $A$ and $B$ snail noodles in yuan.  
$(2)$ Due to the fresh and delicious taste, Xiaoying plans to purchase a total of $30$ boxes of type $A$ and $B$ snail noodles, with the quantity of type $B$ snail noodles not less than the quantity of type $A$ snail noodles. According to an agreement between Xiaoying and the merchant, the unit price of type $A$ snail noodles is reduced by $10$ yuan per box, and the unit price of type $B$ snail noodles is discounted by $10%$ on the original price. How should Xiaoying make the purchase to spend the least amount, and what is the minimum cost in yuan?

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