
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
A certain middle school conducted a survey on students studying advanced math courses. Students who study advanced math courses independently for more than 6 hours per week are called "math enthusiasts", otherwise they are called "non-math enthusiasts". A random sample of 100 people was selected for analysis from the survey results, and the data is shown in the table below:

|         | Math Enthusiasts | Non-Math Enthusiasts | Total |
|---------|------------------|----------------------|-------|
| Male    | 24               | 36                   | 60    |
| Female  | 12               | 28                   | 40    |
| Total   | 36               | 64                   | 100   |

$(1)$ Determine whether there is a 99% confidence that being a "math enthusiast" is related to gender.<br/>$(2)$ Now, from the selected "math enthusiasts", 3 people are selected for a math challenge competition using stratified sampling by gender. It is known that the independent probabilities of male and female students successfully completing the challenge are $\frac{3}{4}$ and $\frac{2}{3}$, respectively. Given that exactly two people successfully complete the challenge, find the probability that a female student successfully completes the challenge.

Reference data and formula:

| $P(K^{2}\geqslant k)$ | 0.10  | 0.05  | 0.010 | 0.001 |
|-----------------------|-------|-------|-------|-------|
| $k$                   | 2.706 | 3.841 | 6.635 | 10.828|

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

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