
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 promotion in 2018. After investigation and calculation, the annual sales volume of the product (i.e., the annual production volume $x$ in ten thousand units) and the annual promotional expenses $m$ (where $m \geqslant 0$) in ten thousand yuan satisfy $x=3- \frac{k}{m+1}$ (where $k$ is a constant). If no promotion is conducted, the annual sales volume of the product will be 1 ten thousand units. It is known that the fixed investment for producing this product in 2018 is 80,000 yuan, and an additional investment of 160,000 yuan is required for every ten thousand units produced. The manufacturer sets the selling price of each unit to 1.5 times the average annual cost per unit (the product cost includes both fixed and additional investments, excluding promotional expenses).

(1) First, find $k$, and express the profit $y$ in ten thousand yuan of the product in 2018 as a function of the annual promotional expenses $m$ in ten thousand yuan.

(2) How much should the manufacturer invest in promotional expenses in 2018 to maximize profit?

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