
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 handicraft factory designed a handicraft with a cost of 11 yuan per piece for trial sales in the market. After market research, it was found that the daily sales volume y (pieces) is a linear function of the selling price x (yuan) (where x is a positive integer). Some corresponding data are shown in the table below:

| Selling Price x (yuan) | ... | 15 | 16 | 17 | 18 | ... |
|------------------------|-----|----|----|----|----|-----|
| Daily Sales Volume y (pieces) | ... | 150 | 140 | 130 | 120 | ... |

(1) Find the analytical expression of y in terms of x;  
(2) If w (yuan) represents the profit obtained by the handicraft factory for selling the handicraft each day, find the analytical expression of w in terms of x;  
(3) At what price per piece will the handicraft factory maximize its daily profit, and what is the maximum profit?

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