
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 the process of building a "two-oriented" society, to promote energy conservation, emission reduction, and the development of a low-carbon economy, a company purchased the production technology of a certain energy-saving product for $800,000. Then, they invested $4,000,000 to purchase production equipment for processing the product. It is known that the cost price of producing and selling this product is $80 per item. According to regulations, the selling price of this product must not be less than $90 per item and not more than $110 per item. The function relationship between the annual sales volume $y$ (in ten thousand items) of this product and the selling price $x$ (in dollars per item) is shown in the table below:

| $x$ (dollars per item) | $90$ | $91$ | ... | $109$ | $110$ |
|------------------------|------|------|-----|-------|-------|
| $y$ (ten thousand items) | $30$ | $29$ | ... | $11$ | $10$ |

$(1)$ The function relationship between $y$ and $x$ is ______, and the range of $x$ is ______;
$(2)$ Find the function relationship between the company's first-year profit $W$ (in ten thousand dollars) and the selling price $x$ (in dollars per item), and determine whether the company is profitable or at a loss in the first year. If profitable, what is the maximum profit? If at a loss, what is the minimum loss?
$(3)$ In the second year, the company decides to donate $2 to local charity for every item sold. If, excluding the maximum profit (or minimum loss) from the first year and the donation in the second year, the total profit for the two years is not less than $2,000,000, how much money can the company donate to local charity in the second year at most by the end of the second year?

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