
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 certain company has developed a high-tech electronic energy-saving product. The company invested $25$ million to purchase a complete set of production equipment. In addition, the cost of producing each product is $20$ yuan, and an annual advertising fee of $5$ million is required. According to regulations, the selling price of the product must not be less than $30$ yuan per item and not more than $70$ yuan per item. The function relationship between the annual sales volume $y$ (in ten thousand units) of the product and the selling price $x$ (yuan per item) is shown in the table below:

| $x$ (yuan per item) | $30$ | $31$ | $\ldots$ | $70$ |
|----------------------|------|------|----------|------|
| $y$ (ten thousand units) | $120$ | $119$ | $\ldots$ | $80$ |

$(1)$ Find the function relationship between $y$ and $x$ and write down the range of values for $x$;  
$(2)$ In the first year, is the company profitable or at a loss? Find the selling price of the product when the profit is maximized or when the loss is minimized;  
$(3)$ Under the premise of $(2)$, that is, when the company maximizes profit or minimizes loss in the first year, the company re-determines the pricing of the product in the second year. Can the company make a total profit of $35$ million yuan over the two years? If yes, find the selling price of the product in the second year; if not, explain the reason.

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