
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 store conducted a market survey and compiled the following information on the price and sales volume of a certain product on the $x$th day $(1\leqslant x\leqslant 90)$ as shown in the table below:

| Time $x$ (days) | $1\leqslant x \lt 50$ | $50\leqslant x\leqslant 90$ |
|-----------------|-----------------------|---------------------------|
| Price (元 per item) | $x+40$ | $90$ |
| Daily sales volume (items) | $200-2x$ | $200-2x$ |

It is known that the cost price of the product is $30$ 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. At which day of selling the product, the profit on that day is maximized, and what is the maximum profit?

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