
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 certain middle school plans to purchase a total of 200 sets of type $A$ and type $B$ desks and chairs. Through bidding, it is found that purchasing one set of type $A$ desks and chairs costs $40$ yuan less than purchasing one set of type $B$ desks and chairs, and that purchasing 3 sets of type $A$ and 5 sets of type $B$ desks and chairs costs a total of $1640$ yuan.  
$(1)$ Find out how much one set of type $A$ desks and chairs and one set of type $B$ desks and chairs cost respectively.  
$(2)$ Based on the actual situation, the school requires that the total cost of purchasing these two types of desks and chairs must not exceed $40880$ yuan, and that the number of type $A$ desks and chairs purchased must not exceed $\frac{2}{3}$ of the number of type $B$ desks and chairs. How many purchasing plans does the school have this time? Which plan results in the lowest total cost? And what is the minimum expenditure?

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