
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]


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Qingming is one of the twenty-four solar terms and also a traditional festival in China. Eating Qingtuan during Qingming Festival is a custom in many places. Before Qingming Festival, the wholesale price of meat floss and salted egg yolk Qingtuan is $10$ yuan cheaper per box than that of sesame Qingtuan. A certain merchant purchased the same number of boxes of sesame Qingtuan for $800$ yuan and meat floss and salted egg yolk Qingtuan for $600$ yuan. During sales, the merchant found that when the selling price of sesame Qingtuan is $50$ yuan per box, 100 boxes can be sold per day. When the price is increased by $1$ yuan per box, 2 fewer boxes are sold per day.<br/>$(1)$ Find the wholesale prices of sesame Qingtuan and meat floss and salted egg yolk Qingtuan;<br/>$(2)$ Given that the selling price of sesame Qingtuan per box does not exceed $65$ yuan, let $W$ represent the profit (in yuan) from selling sesame Qingtuan each day. At what price per box does the merchant maximize the profit from selling sesame Qingtuan each day, and what is the maximum profit?

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