
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 clothing factory provides a certain middle school with the same fabric to produce two different styles of clothing, $A$ and $B$. The amount of fabric used for each set of $A$ clothing is the same, as is the amount used for each set of $B$ clothing. If 1 set of $A$ clothing and 2 sets of $B$ clothing require 5 meters of fabric, and 3 sets of $A$ clothing and 1 set of $B$ clothing require 7 meters of fabric.  
$(1)$ Find out how many meters of fabric are needed for each set of $A$ clothing and each set of $B$ clothing.  
$(2)$ If the middle school needs a total of 100 sets of $A$ and $B$ clothing, and the total amount of fabric used does not exceed 168 meters, then how many sets of $B$ clothing does the factory need to produce at least?

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