
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
Wansui Mountain Great Song Martial Arts City is a large tourist attraction with multiple themes and landscapes that combines Song culture, city wall culture, and seven dynasties culture as its core landscapes, featuring the martial arts culture of the Great Song Dynasty, with a forest natural style and leisure and entertainment functions. The scenic area offers two types of ancient costumes, $A$ and $B$, which are loved by many visitors. Upon investigation, it was found that purchasing 1 piece of type $A$ clothing and 2 pieces of type $B$ clothing costs a total of $110$ yuan; purchasing 2 pieces of type $A$ clothing and 3 pieces of type $B$ clothing costs a total of $190$ yuan.<br/>$(1)$ Find the unit prices of type $A$ and type $B$ clothing respectively;<br/>$(2)$ If the store decides to purchase a total of $100$ pieces of these two types of clothing, with the quantity of type $A$ clothing not less than $\frac{1}{3}$ of the quantity of type $B$ clothing, and in order to promote sales, the store offers a discount of $5$ yuan on each piece of type $A$ clothing. What is the most cost-effective way to purchase type $A$ and type $B$ clothing, and what is the minimum cost?

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