
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]


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In a certain region, all first-year high school students participated in a "Power of Technology" themed knowledge competition. Based on the scoring situation, several first, second, and third prizes were awarded. In order to understand the award situation of students of different genders, 500 first-year high school students participating in the event were randomly selected from the region. The statistical results of the award situation are as follows:

| Gender | Number | Number of First Prize | Number of Second Prize | Number of Third Prize |
|--------|--------|-----------------------|------------------------|-----------------------|
| Male   | 200    | 10                    | 15                     | 15                    |
| Female | 300    | 25                    | 25                     | 40                    |

Assume that the award situation of all students is independent of each other.

$(1)$ From the 200 male students and 300 female students mentioned above, one student is randomly selected from each group. Find the probability that the two selected students both receive the first prize.

$(2)$ Estimate the probability using frequency. One student is randomly selected from the male first-year high school students in the region, and one student is randomly selected from the female first-year high school students in the region. Let $X$ represent the number of award-winning students among these two students. Find the distribution of $X$.

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