
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
In response to the call of the state government's "Enjoy Growth and Bookish Enshi" initiative, the school organized 150 students to participate in a recitation competition. For the event, it was planned to purchase a set of clothing for each student. According to market research, purchasing 1 set of men's clothing and 1 set of women's clothing costs a total of 220 yuan; the cost of purchasing 6 sets of men's clothing is the same as purchasing 5 sets of women's clothing.  
$(1)$ What is the unit price of men's and women's clothing respectively?  
$(2)$ If the number of boys participating in the event does not exceed $\frac{2}{3}$ of the number of girls, and the total cost of purchasing clothing does not exceed 17,000 yuan, how many purchasing plans does the school have? How can the cost be minimized, and what is the minimum cost?

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