
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
Tongwei Hongtai Municipal Corporation sponsors the experimental class of 2016 in our school. The company purchases a small product every month at a factory price of 3 CNY per item. Based on previous statistical data, if the retail price is set to 4 CNY per item, 400 items can be sold each month. For every 0.5 CNY decrease (increase) in the retail price, an additional 40 items can be sold (or fewer by 40 items). All products purchased each month are sold out.
(1) Write down the analytic expression for the function describing the relationship between the selling price $x$ and the profit $y$.
(2) At what selling price per item is the profit maximized? And calculate the maximum profit.

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