
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 high school has 50 students participating in a math competition, with scores (out of 150 points) as follows:

|           | $[90,100)$ | $[100,110)$ | $[110,120)$ | $[120,130)$ | $[130,140)$ | $[140,150]$ |
|-----------|------------|-------------|-------------|-------------|-------------|------------|
| Female    | 1          | 4           | 5           | 5           | 3           | 2          |
| Male      | 0          | 2           | 4           | 12          | 9           | 3          |

$(1)$ If students scoring at least 120 points are considered "mathematically excellent," is there a 95% confidence that being "mathematically excellent" is related to gender?  
$(2)$ If 3 students are randomly selected from male students who scored at least 130 points in the competition, what is the probability that at least 1 of these 3 students scored in the range $[140,150]$?  
Given: $K^2 = \frac{n(ad-bc)^2}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.

| $P(K^{2}\geqslant k_{0})$ | 0.05 | 0.025 | 0.010 | 0.01  |
|---------------------------|------|-------|-------|-------|
| $k_{0}$                   | 3.841| 5.024 | 6.635 | 10.828|

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