
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

...

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system
Soybeans are an important crop in China with a long history of cultivation. A certain experimental base has bred a new variety of soybeans to test the optimal sowing date. Experiments were conducted on two test fields, $A$ and $B$ (with the same soil conditions). The new soybean variety was sown on June 25th in field $A$ and on July 10th in field $B$. When harvesting the soybeans, 20 samples were randomly taken from each field (each sample containing 1,000 grains) and the weight of each sample was measured (in grams). The weights were grouped into intervals $\left[100,150\right)$, $\left[150,200\right)$, and $\left[200,250\right]$, with the following distribution:<br/>

|            | $\left[100,150\right)$ | $\left[150,200\right)$ | $\left[200,250\right]$ |
|------------|-------------------------|-------------------------|-----------------------|
| Field $A$  | 3                       | 6                       | 11                    |
| Field $B$  | 6                       | 10                      | 4                     |

Grains with a weight of at least 200 grams are considered full, otherwise considered not full.<br/>
$(1)$ Based on a significance level of $a=0.025$, conduct an independence test to analyze whether the fullness of soybean grains is related to the sowing date.<br/>
$(2)$ Take one sample from each of fields $A$ and $B$, and calculate the probability of having at least one full grain among the samples.<br/>
$(3)$ Using the sample to estimate the population, randomly select 100 samples of soybeans from field $A$, denoting the number of full grains as $X$. Calculate the mathematical expectation and variance of $X$.<br/>
Reference formula: ${\chi}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.<br/>

| $\alpha $ | 0.15 | 0.10 | 0.05 | 0.025 | 0.010 | 0.001 |
|-----------|------|------|------|-------|-------|-------|
| $x_{0}$   | 2.072| 2.706| 3.841| 5.024 | 6.635 | 10.828|

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