
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 school wants to purchase two types of notebooks, expecting to spend $460. The first type of notebook costs $3 each, and the second type costs $2 each. The school plans to buy 20 more notebooks of the first type than twice the number of notebooks of the second type.  
$(1)$ Find the quantities of the two types of notebooks purchased.  
$(2)$ A store is having a promotion where the first type of notebook is sold at an 20% discount and the second type at a 10% discount. How much money can the school save this time?

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