
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 ninth-grade math interest group conducted a market survey and compiled the relevant information on the price and sales volume of a certain product for $x$ days $(1\leqslant x\leqslant 70$ and $x$ is an integer) as shown in the table below:

| Time $x$ (days) | $1\leqslant x \lt 40$ | $40\leqslant x\leqslant 70$ |
|-----------------|-----------------------|---------------------------|
| Price (元 per item) | $x+45$ | $85$ |
| Daily Sales (items) | $150-2x$ | $150-2x$ |

Given that the cost price of the product is $30$ yuan per item, let the daily profit from selling the product be $y$ yuan.
$(1)$ Find the functional relationship between $y$ and $x$;
$(2)$ At which day of selling the product, the daily profit is maximized, and what is the maximum profit?
$(3)$ How many days during the selling process does the daily profit not fall below $3250$ yuan? Please write down the results directly.

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