
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 save energy and reduce emissions, a certain school in our city is preparing to purchase a certain brand of energy-saving lamps. It is known that $3$ of type $A$ energy-saving lamps and $5$ of type $B$ energy-saving lamps cost a total of $50$ yuan, while $1$ of type $A$ energy-saving lamp and $3$ of type $B$ energy-saving lamps cost $26$ yuan.  
$(1)$ Find the price of $1$ type $A$ energy-saving lamp and $1$ type $B$ energy-saving lamp respectively.  
$(2)$ The school plans to purchase a total of $200$ lamps of these two types. It is required to purchase $a$ type $A$ lamps. Let the total cost of purchasing the two types of lamps be $W$ yuan.  
① Find the functional relationship between $W$ and $a$.  
② When $a=80$, find the total cost of purchasing the two types of lamps.

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