
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 31st Universiade, "Chengdu Achieves Dreams," will open on June 26, 2022. A souvenir of the Universiade named "Rong Bao," with a cost of 30 yuan, has been put on the market for trial sales by merchants. According to market research, the daily sales volume $y$ (pieces) is a linear function of the selling price $x$ (33 ≤ x ≤ 58, and x is a positive integer) of each item. Some corresponding data is shown in the table below:

| Selling Price $x$ (yuan) | ... | 35 | 36 | 37 | 38 | ... |
|--------------------------|-----|----|----|----|----|-----|
| Daily Sales Volume $y$ (pieces) | ... | 90 | 88 | 86 | 84 | ... |

$(1)$ Write down the functional relationship between $y$ and $x$ directly.

$(2)$ Find the selling price per item when the merchant earns 800 yuan per day.

$(3)$ Determine the selling price of the souvenir when the merchant maximizes the profit obtained from selling the souvenir each day. What is the maximum profit?

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