
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
In the "Mathematics in Daily Life" exploration activity, Xiaoming conducted a market survey to study the relationship between the selling price, sales volume, and profit of a certain product. Xiaoming organized the relevant data of the product as shown in the table below.

| Time $x$ (days) | $1\leqslant x \lt 30$ | $30\leqslant x\leqslant 50$ |
|-----------------|-----------------------|---------------------------|
| Selling Price (元/item) | $x+40$ | $70$ |
| Daily Sales Volume (items) | $100-2x$ | - |

It is known that the cost price of the product is $10$ yuan per item. Let $y$ yuan be the profit from selling the product each day.
- $(1)$ Find the functional relationship between $y$ and $x$.
- $(2)$ On which day of selling the product is the profit maximized, and what is the maximum profit?

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