
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
Students majoring in "Plant Nutrition" at a certain university chose the increase in plant height of cockscomb as the research object to observe the effects of long-acting fertilizer and slow-release fertilizer on crops. The three treatments of long-acting fertilizer, slow-release fertilizer, and no fertilizer correspond to three groups labeled as $1$, $2$, and $3$. After observing for a period of time, $40$ cockscomb plants were randomly selected from each of the three groups as samples, and the corresponding data on plant height increase were organized as shown in the table below.

| Plant Height Increase (cm) | $(4,7]$ | $(7,10]$ | $(10,13]$ | $(13,16]$ |
|----------------------------|---------|----------|-----------|-----------|
| Group 1 cockscomb count     | $9$     | $20$     | $9$       | $2$       |
| Group 2 cockscomb count     | $4$     | $16$     | $16$      | $4$       |
| Group 3 cockscomb count     | $13$    | $12$     | $13$      | $2$       |

Assuming frequency estimates probability, and all cockscomb growth conditions are independent of each other.

$(Ⅰ)$ From all cockscomb plants in Group 1, one plant is randomly selected. Estimate the probability that the plant height increase is in the range $(7,10]$ cm.

$(Ⅱ)$ From all cockscomb plants in Group 1, Group 2, and Group 3, one plant is randomly selected from each group. Let $X$ be the number of plants among these three selected plants with a plant height increase in the range $(7,10]$ cm. Find the probability distribution of $X$ and the mathematical expectation $E(X)$.

$(Ⅲ)$ Use "$\xi _{k}=1$" to represent that the plant height increase of Group $k$ cockscomb is in the range $(4,10]$, and "$\xi _{k}=0$" to represent that the plant height increase of Group $k$ cockscomb is in the range $(10,16]$ cm, where $k=1$, $2$, $3$. Write down the relationship of variances $D(\xi _{1})$, $D(\xi _{2})$, $D(\xi _{3})$ directly without requiring proof.

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