
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 factory, in order to determine the total production of product $A$ in the year $2024$, retrieved the corresponding relationship between the total production $x$ (where $x \gt 0$) of product $A$ and the total cost $y$ in the years from $2020$ to $2023$ (as shown in the table below) as the basis for establishing the function relationship between $y$ and $x, thereby achieving estimation and prediction. The factory refers to this function as the "reference function."

| Total Production of Product $A$ $x$ (in 10,000 units) | 1   | 2   | 3   | 4   |
|-------------------------------------------------------|-----|-----|-----|-----|
| Total Cost $y$ (in 10,000 yuan)                       | 12  | 17  | 25  | 32  |

The factory plans to use the following three function expressions as alternative "reference functions": ① $y=5x+7$; ② $y=x^{2}+2x+9$; ③ $y=2.5\times 2^{x}+7$.

$(1)$ Which function expression should the factory choose as the most reasonable "reference function"? Please explain the reason.

$(2)$ Based on the selected "reference function" in $(1)$, when the factory estimates the production of product $A$ in the year $2024$, at what production quantity will the unit cost (i.e., total cost divided by total production) be the lowest? Calculate this lowest unit cost.

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