
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
In order to increase the annual production of products, a company plans to carry out technological reforms in 2013. After investigation and calculation, the annual production $x$ (in ten thousand units) and the cost of technological reform $m$ (in ten thousand yuan) $(m \geqslant 0)$ satisfy $x=3- \frac{k}{m+1}$ (where $k$ is a constant). If no technological reform is undertaken, the annual production of the product will only be 1 ten thousand units. It is known that the fixed income from producing this product in 2013 is 80,000 yuan, and an additional investment of 160,000 yuan is required for every ten thousand units produced. Due to favorable market conditions, all products produced by the factory can be sold. The factory sets the selling price of each product at 1.5 times the production cost (the production cost includes both fixed and additional investments).

(Ⅰ) Determine the value of $k$, and express the profit $y$ (in ten thousand yuan) in 2013 as a function of the technological reform cost $m$ (in ten thousand yuan) (profit = sales amount - production cost - technological reform cost).

(Ⅱ) How much should the company invest in technological reform in 2013 to maximize the factory's profit?

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