
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]


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A company in Xiangyang city proactively responded to the government's call for "innovative development" and developed a new product. It's known that the cost of research and development, and the production of this product is 30 yuan per unit. The annual sales volume $y$ (in ten thousand units) is a function of the selling price $x$ (in yuan per unit), given by the piecewise function:
$$y= \begin{cases} -2x+140, & \text{for } 40\leqslant x < 60\\ -x+80, & \text{for } 60\leqslant x\leqslant 70 \end{cases}.$$
(1) If the company's annual profit from selling this product is $W$ (in ten thousand yuan), write the function defining the annual profit $W$ in terms of the selling price $x$ (in yuan per unit).
(2) When the selling price $x$ (in yuan per unit) is set to what value does the company obtain the maximum annual profit? What is this maximum annual profit?
(3) If the company's annual profit from selling this product is not less than 750 ten thousand yuan, determine the range of selling price $x$ (in yuan per unit).

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