
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 Hangzhou, a research group randomly selected 100 citizens in our city for a survey and obtained the following data:

| Weekly Exercise Frequency | 1 time | 2 times | 3 times | 4 times | 5 times | 6 times and above |
|---------------------------|--------|---------|---------|---------|---------|-------------------|
| Male                      | 4      | 6       | 5       | 3       | 4       | 28                |
| Female                    | 7      | 5       | 8       | 7       | 6       | 17                |

$(1)$ If it is considered that users who exercise 4 times or more per week are "fitness enthusiasts," please complete a $2 \times 2$ contingency table. Based on the small probability value $\alpha = 0.05$, conduct an independence test to determine whether "fitness enthusiasts" and "gender" are related.

$(2)$ Assuming that Xiao Hong, a citizen of Hangzhou, has a probability of $\frac{2}{11}$ of going to gym A for the first time, and a probability of $\frac{9}{11}$ of going to gym B. From the second time onwards, if she went to gym A the previous time, the probability of not going to A this time is $\frac{1}{4}$; if she went to gym B the previous time, the probability of not going to A this time is $\frac{1}{3}$. Let $P_{n}$ denote the probability of going to gym A for the $n$th time. Which gym is more likely to be visited for the 10th time?

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}
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