
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
Light food is a form of catering, where "light" not only refers to the quantity of ingredients, but also to the simple cooking methods that retain the original nutrition and flavor of the ingredients. In recent years, with the increasing health awareness of consumers and the popularity of the beauty economy, the light food industry has experienced rapid development. A certain media company conducted a sample survey of Chinese light food consumers to obtain consumer behavior data. The frequency distribution table below shows the frequency of consuming light food among 400 Chinese light food consumers (with 100 people in each of the 4 age groups).

| Frequency | $\left[12,25\right)$ | $\left[25,38\right)$ | $\left[38,51\right)$ | $\left[51,64\right]$ |
|-----------|----------------------|----------------------|----------------------|----------------------|
| Occasionally 1 time | 30 | 15 | 5 | 10 |
| 1-3 times per week | 40 | 40 | 30 | 50 |
| 4-6 times per week | 25 | 40 | 45 | 30 |
| 1 or more times per day | 5 | 5 | 20 | 10 |

$(1)$ If consumers aged between $\left[12,38\right)$ are called teenagers and those aged between $\left[38,64\right]$ are called middle-aged and elderly, and those who consume light food no more than 3 times a week are considered to have a low frequency of consuming light food, while those who consume light food 4 or more times a week are considered to have a high frequency of consuming light food, determine whether the frequency of consuming light food is related to age based on an independence test with a small probability value of $\alpha = 0.01$.

$(2)$ From the sample consumers who consume light food 1 or more times per day, stratified random sampling is conducted based on the age groups in the table. 8 people are selected proportionally, and then 3 people are randomly selected from these 8 people. Let the number of people aged between $\left[25,38\right)$ and $\left[51,64\right]$ among these 3 people be $X$, $Y$, and $\xi = |X-Y|$. Find the distribution table and expectation of $\xi$.

$(3)$ It is known that Xiao Li consumes light food for both breakfast and dinner every day, and for breakfast and dinner, she randomly chooses one of the three types of light food: low-calorie dessert, whole wheat sandwich, or fruit and vegetable juice. If Xiao Li chooses low-calorie dessert, whole wheat sandwich, or fruit and vegetable juice for breakfast, the probabilities of choosing low-calorie dessert for dinner are $\frac{1}{5}$, $\frac{2}{5}$, and $\frac{2}{3}$, respectively. Find the probability that Xiao Li chooses low-calorie dessert for dinner.

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

Given table:

| $\alpha$ | 0.10 | 0.05 | 0.01 | 0.005 | 0.001 |
|----------|------|------|------|-------|-------|
| $x_{\alpha}$ | 2.706 | 3.841 | 6.635 | 7.879 | 10.828 |

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