
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
During the "Double Eleven" promotion on Taobao, a handicraft item is sold with a cost of $50$ yuan per piece. It was found during the sales period that when the selling price is $100$ yuan, the daily sales volume is 50 pieces. For every $1$ yuan decrease in the selling price, an additional 5 pieces can be sold per day, but the selling price must not be lower than the cost. Let $x$ represent the selling price in yuan and $y$ represent the daily profit in yuan.<br/>$(1)$ Find the function expression between $y$ and $x$;<br/>$(2)$ Determine at what selling price the daily profit is maximized and what the maximum profit is;<br/>$(3)$ If the daily profit must not be less than $4000$ yuan, then the minimum total cost per day should be ______ yuan? (Total daily cost $=$ Cost per piece $\times$ Daily sales volume)

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