
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
A certain scenic spot shop sells a kind of souvenir, with a purchase price of $40$ yuan per item. According to market research, when the selling price of the souvenir is $50$ yuan per item, 200 items can be sold per day. When the selling price increases by $1$ yuan per item, the daily sales quantity decreases by 10 items.<br/>$(1)$ When the selling price per item is $52$ yuan, the daily sales quantity of the souvenir is ______ items;<br/>$(2)$ For what value of the selling price per item $x$ will the profit $y$ obtained from selling the souvenir be maximized per day? Also, calculate the maximum profit.

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