
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
National fitness is the foundation and guarantee for all people to enhance their physical fitness and live a healthy life. In order to study the fitness situation of citizens in Fuzhou, a research group randomly selected 100 citizens in our city for a survey and obtained the following data in the table:

| Weekly Exercise Frequency | 1 time | 2 times | 3 times | 4 times | 5 times | 6 times and above |
|---------------------------|--------|---------|---------|---------|---------|-------------------|
| Male                      | 4      | 3       | 3       | 3       | 7       | 30                |
| Female                    | 6      | 5       | 4       | 7       | 8       | 20                |

$(1)$ If users who exercise more than 3 times a week are considered "fitness enthusiasts," please complete the $2 \times 2$ contingency table (see answer sheet) and, based on the small probability value $\alpha = 0.05$, conduct an independence test to determine whether "fitness enthusiasts" and "gender" are related.  
$(2)$ Users who exercise 6 times or more per week are called "fitness experts." Let the random variable $Y$ be the total amount of rewards. In a yoga studio in Fuzhou, each selected female "fitness expert" will be rewarded with a $1000$ yuan fitness card. Calculate the mathematical expectation of $Y$.  
Given: ${K}^{2} = \frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$,  
| $\alpha$ | 0.10 | 0.05 | 0.01 | 0.005 | 0.001 |
|----------|------|------|------|-------|-------|
| $k_{\alpha}$ | 2.706 | 3.841 | 6.635 | 7.879 | 10.828 |

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