
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
"Chen Chen", "Cong Cong", and "Lian Lian" are the mascots of the 2023 Hangzhou Asian Games, collectively named "Jiangnan Yi". They are derived from the famous line by the Tang Dynasty poet Bai Juyi, "Remembering Jiangnan, most memorable is Hangzhou", which integrates the historical, cultural, natural, and innovative genes of Hangzhou. A Chinese company can produce the mascots "Chen Chen", "Cong Cong", and "Lian Lian" for the Hangzhou Asian Games. According to market research and forecasts, the relationship between investment cost $x$ (in hundreds of million yuan) and profit $y$ (in hundreds of million yuan) is as shown in the table below:

| $x$ (hundreds of million yuan) | $\cdots$ | $2$ | $\cdots$ | $4$ | $\cdots$ | $12$ | $\cdots$ |
|--------------------------------|----------|-----|----------|-----|----------|------|----------|
| $y$ (hundreds of million yuan) | $\cdots$ | $0.4$ | $\cdots$ | $0.8$ | $\cdots$ | $12.8$ | $\cdots$ |

When the investment cost $x$ is not more than $12$ hundred million yuan, the relationship between profit $y$ (in hundred million yuan) and investment cost $x$ (in hundred million yuan) has two function models to choose from: $y = Ax^{2} + B$ ($A > 0$) and $y = Ta^{x}$ ($T > 0, a > 1$).

$(1)$ When the investment cost $x$ is not more than $12$ hundred million yuan, select the function model that you think is most realistic and find the corresponding function expression.

$(2)$ When the investment cost $x$ is more than $12$ hundred million yuan, the profit $y$ (in hundred million yuan) and investment cost $x$ (in hundred million yuan) satisfy the relationship $y = -0.2(x-12)(x-17) + 12.8$. Combining the result of question (1), to achieve a profit of at least one billion yuan, in what range should the investment cost $x$ (in hundred million yuan) be controlled? (Round the result to one decimal place) (Reference data: $\lg 2 \approx 0.30$)

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