
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
The owner of a clothing store went to the manufacturer to purchase clothing of two brands, $A$ and $B$. If they buy 3 sets of brand $A$ clothing and 2 sets of brand $B$ clothing, it costs $450$ yuan. If they buy 1 set of brand $A$ clothing and 1 set of brand $B$ clothing, it costs $175$ yuan.  
$(1)$ Find the cost price per set of brand $A$ and brand $B$ clothing.  
$(2)$ If selling 1 set of brand $A$ clothing earns a profit of $30$ yuan, and selling 1 set of brand $B$ clothing earns a profit of $20$ yuan, the owner of the clothing store decides to purchase a total of 100 sets of brand $A$ and brand $B$ clothing. After selling all the clothing, the total profit will be no less than $2400$ yuan. How many sets of brand $A$ clothing should be purchased at least?

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