
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
The red lantern symbolizes family reunion and prosperity, hanging lanterns has become a traditional culture in our country. Xiao Ming purchased two types of red lanterns, A and B, before the Spring Festival. The quantity of lanterns purchased for type A at $3120$ yuan is the same as the quantity purchased for type B at $4200$ yuan. It is known that the cost of each pair of type B lanterns is $9$ yuan more than the cost of each pair of type A lanterns.<br/>$(1)$ Find the cost of each pair of type A and type B lanterns;<br/>$(2)$ After a market survey, it was found that when the selling price of each pair of type B lanterns is $50$ yuan, 98 pairs can be sold per day. For each increase of $1$ yuan in selling price, 2 pairs less are sold per day. The price department stipulates that the selling price should not exceed $65$ yuan per pair. If the selling price of type B lanterns increases by $x$ yuan, Xiao Ming will earn a profit of $y$ yuan per day.<br/>① Find the analytical expression of the function between $y$ and $x$;<br/>② At what selling price per pair of type B lanterns will the profit be maximized per day? What is the maximum profit?

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