
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
To prepare for the inspection of the "National Civilized and Sanitary City," the sanitation department of a certain city is ready to purchase two models of garbage bins, $A$ and $B$. Through market research, it was found that purchasing $1$ type $A$ garbage bin and $2$ type $B$ garbage bins costs a total of $340$ yuan, and purchasing $3$ type $A$ garbage bins and $1$ type $B$ garbage bin costs a total of $420$ yuan.  
$(1)$ Find the prices of $1$ type $A$ garbage bin and $1$ type $B$ garbage bin.  
$(2)$ The city now needs to purchase a total of $30$ garbage bins of types $A$ and $B, with $x$ ($x\leqslant 16$) type $A$ garbage bins.  
① Find the functional expression of the total cost $W$ (in yuan) of purchasing garbage bins in terms of the number of type $A$ garbage bins $x$.  
② For how many type $A$ garbage bins should be purchased to minimize the total cost? What is the minimum cost?

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