
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 manufacturer plans to hold a sales promotion event. After investigations and calculations, it was found that the annual sales volume (x in ten thousand units) of the product correlates with the annual promotional expenses (m in ten thousand yuan, m≥0) in the following way: $x=3-\frac{k}{m+1}$ (k is a constant). Without any promotional activities, the annual sales volume would only be 1 ten thousand units. It's known that in 2004, the fixed investment for producing this product was 80 thousand yuan, and an additional investment of 160 thousand yuan is needed for every ten thousand units produced. The manufacturer sets the sales price of each product at 1.5 times the average annual cost per product (including both fixed and additional investments).

(1) Express the manufacturer's profit (y in ten thousand yuan) as a function of the annual promotional expenses (m in ten thousand yuan).
(2) How much should the manufacturer invest in promotion expenses in order to maximize profits?

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