
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
Badminton sports have the significance of striving, progress, and positivity, and also require athletes to have carefulness and quick agility. In order to understand whether students at a certain university are interested in badminton sports, the badminton sports association of the university randomly selected 300 students for a survey. The ratio of male to female students is 2:1, with 80% of female students interested in badminton sports, while 30 male students expressed no interest in badminton sports.  
$(1)$ Complete the $2\times 2$ contingency table. Based on the small probability value $\alpha =0.1$ of the independence test, can it be concluded that "interest in badminton sports is related to gender"?  
$(2)$ In order to increase students' participation in badminton sports, the school held a badminton competition. The competition is divided into two stages, with the first stage adopting a single round-robin format, where each match follows a best-of-three games system. The students who enter the second stage of the competition are selected based on their points. The points system for each match is as follows: students who win by $2:0$ earn 3 points, while those who lose earn 0 points; students who win by $2:1$ earn 2 points, and those who lose earn 1 point. Among the matches, the match between students Xiao Qiang and Xiao Ming received much attention. If the probability of Xiao Qiang winning each game is $p=\frac{3}{5}$, and the points earned by Xiao Qiang are denoted as $X$, find the distribution table and expectation of $X$.  
Given: $\chi ^{2}=\frac{n(ad-bc)^2}{(a+b)(c+d)(a+c)(a+d)}$, where $n=a+b+c+d$.  
$\alpha$ | $0.5$ | $0.40$ | $0.25$ | $0.150$ | $0.100$ | $0.050$
--- | --- | --- | --- | --- | --- | ---
$x_{\alpha}$ | $0.455$ | $0.780$ | $1.323$ | $2.072$ | $2.706$ | $3.841$

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