
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
In order to enhance students' interest in learning mathematics, a certain middle school is conducting a mathematics activity with the theme "Feel the Charm of Mathematics, Enjoy the Fun of Mathematics." The school plans to purchase two types of prizes, $A$ and $B$, to reward students who excel in the mathematics activity. It is known that purchasing $1$ prize of type $A$ and $2$ prizes of type $B$ costs a total of $64$ dollars, and purchasing $2$ prizes of type $A$ and $1$ prize of type $B$ costs a total of $56$ dollars.  
$(1)$ What is the price of each prize of type $A$ and type $B$?  
$(2)$ Based on the needs, the school plans to purchase a total of $80$ prizes of types $A$ and $B$. Let $a$ be the number of prizes of type $A$ purchased, and the total cost is $w$ dollars. Find the functional relationship between $w$ and $a$, and directly state the range of values for $a$.  
$(3)$ Under the conditions of $(2)$, if it is required that the number of prizes of type $A$ purchased does not exceed three times the number of prizes of type $B$, find the minimum total cost required.

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