
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 supermarket purchases two types of goods, A and B. It costs 6 yuan more to buy one A item than to buy one B item, and the amount spent on purchasing A items for 400 yuan is equal to the amount spent on purchasing B items for 100 yuan.  
$(1)$ Find out how much it costs to buy one A item and one B item each.  
$(2)$ The store plans to purchase a total of 40 A and B items, and requires the number of items to be positive integers. If the quantity of A items is not less than three times the quantity of B items, and the total cost of purchasing A and B items is not less than 230 yuan and not more than 266 yuan, how many purchasing plans does the supermarket have? Which plan costs less?

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