
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 full score for this question is 12 points.) A company designed a craft, with each item costing 50 yuan to produce. To set a reasonable price, the product was put on the market for a trial sale. According to market research, when the selling price is 100 yuan, the daily sales volume is 50 items, and for every 1 yuan decrease in the selling price, 5 more items can be sold per day, but the selling price must not be lower than the cost.

(1) Find the functional relationship between the daily sales profit $y$ (in yuan) and the selling price $x$ (in yuan);

(2) At what selling price is the daily sales profit maximized? What is the maximum profit?

(3) If the company wants to ensure that the daily sales profit is not less than 4000 yuan, and the total daily cost does not exceed 7000 yuan, within what range should the selling price be controlled? (The total daily cost = cost per item × daily sales volume)

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