
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
Whether it is the international situation or the domestic consumption situation, the year $2023$ is a challenging year. In order to cope with the complex economic situation, various regions have introduced various policies to promote economic development, actively responding to the current economic situation and achieving good results. In order to understand the impact of the mass media on this promotion activity, a certain city randomly selected $6$ large retail stores in the city and obtained data on their advertising costs $x$ (in units of ten thousand yuan) and sales volume $y$ (in units of ten thousand yuan) as shown in the table below:

| Store | 1 | 2 | 3 | 4 | 5 | 6 |
|-------|---|---|---|---|---|---|
| Advertising Cost | 2 | 3 | 5 | 6 | 8 | 12 |
| Sales Volume | 30 | 34 | 40 | 45 | 50 | 60 |

$(1)$ Find the linear regression equation of $y$ with respect to $x$ and predict when the advertising cost is at least how many ten thousand yuan (rounded to the nearest integer), the sales volume can exceed $100$ thousand yuan.

$(2)$ In economic activities, people often pay attention to the input-output ratio. In this promotion activity, the ratio of sales volume to the input advertising cost is denoted as $\lambda$. If $\lambda \geq 9$, this advertising plan is considered efficient; otherwise, it is considered inefficient. Three stores are randomly selected from these $6$ stores. Find the probability that at least one of these $3$ stores has an efficient advertising plan.

Given: $\sum_{i=1}^{6}x_iy_i=1752$, the regression line equation $\hat{y}=\hat{a}+\hat{b}x$ where the least squares estimates of $\hat{b}$ and $\hat{a}$ are $\hat{b}=\frac{\sum_{i=1}^{n}x_iy_i-n\bar{x}\bar{y}}{\sum_{i=1}^{n}x_i^2-n\bar{x}^2}$ and $\hat{a}=\bar{y}-\hat{b}\bar{x}$.

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