
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 school conducts physical fitness tests for students every year, including tests for lung capacity, a 50-meter run, standing long jump, and other items. The lung capacity levels (excellent, good, qualified, unqualified) of 80 male students at the school are statistically analyzed. The results are shown in the following contingency table:

| Height | Lung Capacity Level | Total |
|--------|----------------------|-------|
| Below 175 cm | 22 | 22 | 44 |
| 175 cm or above | 30 | 6 | 36 |
| Total | 52 | 28 | 80 |

$(1)$ Can we be $99.5\%$ confident that there is an association between the height of male students and their lung capacity levels?

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

| $P(K^{2}\geqslant k)$ | 0.01 | 0.005 | 0.001 |
|-----------------------|------|-------|-------|
| $k$ | 6.635 | 7.879 | 10.828 |

$(2)$ A fitness test group consists of 6 male students, including 1 unqualified, 4 good, and 1 excellent lung capacity levels. The scoring system is as follows: unqualified - 0 points, qualified - 1 point, good - 2 points, excellent - 3 points. Two students are randomly selected from the group. Let the sum of their lung capacity level scores be $X$. Find the probability distribution and mean of $X$.

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