
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 2019, the 5G base station officially obtained the approval of the Ministry of Industry and Information Technology, marking the beginning of the "5G era" in China. Compared to 4G, 5G features "higher network speed, low latency, high reliability, low power consumption, and massive connectivity." Recently, this technology has been widely applied in various fields such as industry, energy, and education. To understand the satisfaction level of customers with the 5G network, a certain operator selected 300 customers from the operation system for a survey. Among them, the ratio of young customers (≤45 years old) to middle-aged and elderly customers (≥45 years old) is 3:2. Among the 220 customers who are satisfied, the ratio of middle-aged and elderly customers is 4/11.

$(1)$ Complete the $2 \times 2$ contingency table. Based on the independence test with a small probability value of $\alpha = 0.05$, can it be concluded that there is an association between the satisfaction with the 5G network and age?

\[
\begin{array}{|c|c|c|c|}
\hline
\text{Age} & \text{Satisfaction with 5G Network} & \text{Total} \\
\hline
\text{Young} (\leq 45 \text{ years old}) & & & \\
\hline
\text{Middle-aged and Elderly} (> 45 \text{ years old}) & & & \\
\hline
\text{Total} & & & \\
\hline
\end{array}
\]

$(2)$ To better promote the 5G network, the operator plans to conduct a lottery. The rules are as follows: Participants randomly draw a ball from a box containing 3 red balls and 6 white balls (same shape, size, and texture). After drawing, the ball is put back. Drawing a red ball rewards a $5$ yuan recharge voucher, while drawing a white ball rewards a $2$ yuan recharge voucher. If 100 customers are planned to participate in the lottery, calculate the mathematical expectation $X$ of the total amount of recharge vouchers that the operator needs to provide.

Given: $\chi ^{2}=\frac{n(ad-bc)^2}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.

\[
\begin{array}{|c|c|c|c|}
\hline
P(\chi ^{2} \geq k) & 0.01 & 0.05 & 0.025 \\
\hline
k & 2.706 & 3.841 & 5.024 \\
\hline
\end{array}
\]

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