
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
The CCTV "National Brand Plan" program team conducted a survey on the brand promotion of new energy vehicles using an online platform for people aged between 20 and 60 years. They randomly selected 600 people from the survey participants and divided them into two categories: those who are more interested and less interested in new energy vehicles. The distribution is shown in the table below:

| Age/years | $[20,30)$ | $[30,40)$ | $[40,50)$ | $[50,60]$ |
|-----------|-----------|-----------|-----------|-----------|
| Gender    | Male      | Female    | Male      | Female    | Male      | Female    | Male      | Female    |
| Number    | $40$      | $10$      | $120$     | $70$      | $160$     | $100$     | $80$      | $20$      |
| Proportion of Interest | $20\%$ | $50\%$ | $60\%$ | $70\%$ | $70\%$ | $80\%$ | $60\%$ | $80\%$ |

$(1)$ Fill in the contingency table and determine whether gender is related to the interest in new energy vehicles with a probability of error not exceeding $0.01$.

|         | Interested | Less Interested | Total |
|---------|------------|-----------------|-------|
| Male    | $240$      | $160$           | $400$ |
| Female  | $150$      | $50$            | $200$ |
| Total   | $390$      | $210$           | $600$ |

Calculate $K^{2}=\dfrac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}=\dfrac{600(240\times 50-160\times 150)^{2}}{400\times 200\times 390\times 210}\approx 13.19 > 6.635$. Therefore, with $99\%$ confidence, gender is related to the interest in new energy vehicles.

$(2)$ Based on $(1)$, the male-to-female ratio is $2:1$. From the 6 selected individuals, 2 are female and 4 are male. Let $X$ be the number of females among the 3 individuals selected for the TV show. Find the distribution and expectation of $X$.

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