
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
Table 1: The number of "excellent" students with "standard actions" tested<br/>
<table>
<thead>
<tr>
<th>Group Number</th>
<th>1</th>
<th>2</th>
<th>3</th>
<th>4</th>
<th>5</th>
</tr>
</thead>
<tbody>
<tr>
<td>"Excellent" Count</td>
<td>1</td>
<td>2</td>
<td>4</td>
<td>6</td>
<td>7</td>
</tr>
</tbody>
</table>

Table 2: The number of "excellent" students with "non-standard actions" tested<br/>
<table>
<thead>
<tr>
<th>Group Number</th>
<th>1</th>
<th>2</th>
<th>3</th>
<th>4</th>
<th>5</th>
</tr>
</thead>
<tbody>
<tr>
<td>"Excellent" Count</td>
<td>0</td>
<td>1</td>
<td>2</td>
<td>3</td>
<td>4</td>
</tr>
</tbody>
</table>

$(1)$ Fill in the following table. Based on the small probability value $\alpha = 0.05$, conduct an independence test analysis to infer whether "excellent" and "average" are related to "standard actions" during training.<br/>
<table>
<thead>
<tr>
<th></th>
<th>"Excellent"</th>
<th>"Average"</th>
<th>Total</th>
</tr>
<tr>
<td>"Standard Actions"</td>
<td>20</td>
<td>30</td>
<td>50</td>
</tr>
<tr>
<td>"Non-standard Actions"</td>
<td>10</td>
<td>40</td>
<td>50</td>
</tr>
<tr>
<td>Total</td>
<td>30</td>
<td>70</td>
<td>100</td>
</tr>
</thead>
</table>

$(2)$ In the test results of students with "standard actions," let $x$ represent the group number and $y$ represent the number of "excellent" students. Calculate the average values $\overline{x}$ and $\overline{y}$ from Table 1 and the empirical regression equation of $y$ with respect to $x$: $\hat{y} = \hat{b}x + \hat{a}$.<br/>
Reference data and formula: ${\chi}^{2} = \frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$, $n = a+b+c+d$.<br/>
<table>
<thead>
<tr>
<th>$\alpha$</th>
<th>0.15</th>
<th>0.10</th>
<th>0.05</th>
<th>0.025</th>
<th>0.010</th>
<th>0.005</th>
<th>0.001</th>
</tr>
<tr>
<th>$x_{\alpha}$</th>
<th>2.072</th>
<th>2.706</th>
<th>3.841</th>
<th>5.024</th>
<th>6.635</th>
<th>7.879</th>
<th>10.828</th>
</tr>
</thead>
</table>
$\sum_{i=1}^{5}{{x}_{i}}{y}_{i} = 76$, $\sum_{i=1}^{5}{{x}_{i}^{2}} = 55$, $\hat{b} = \frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}-b\overline{xy}}{\sum_{i=1}^{n}{{x}_{i}}^{2}-n{\overline{x}}^{2}}$, $\hat{a} = \overline{y}-\hat{b}\overline{x}$.

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