
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
A self-limiting disease refers to a disease with the characteristics of self-relief and self-dissipation. It is known that a certain self-limiting disease can generally recover after $10$ days without medication. To study the effect of drug $A$ on this self-limiting disease, a research institute conducted a double-blind experiment, randomly dividing $100$ volunteers who were initially diagnosed with the disease into two groups. Group A used drug $A$ normally, while group B used a placebo instead of medication. The following $2\times 2$ contingency table was obtained through statistics:

|          | Recovered in less than $10$ days | Recovered after $10$ days | Total |
|----------|----------------------------------|---------------------------|-------|
| Group A  | $30$                             | $20$                      | $50$  |
| Group B  | $10$                             | $40$                      | $50$  |
| Total    | $40$                             | $60$                      | $100$ |

$(1)$ Based on the data provided in the $2\times 2$ contingency table, can we be $99\%$ confident that drug $A$ is related to recovery in less than $10$ days?  
$(2)$ If the frequency of recovery after $10$ days in Group A is considered as the probability of drug $A$ being ineffective, and now $4$ people are randomly selected from those who have the disease and have used drug $A$, let $X$ be the number of people for whom drug $A$ is ineffective. Find the distribution table and the expected value of $X$.  
Attachment: ${K^2}=\frac{{n{{({ad-bc})}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$, $n=a+b+c+d$.  
| $P(K^{2}\geqslant k_{0})$ | $0.100$ | $0.050$ | $0.025$ | $0.010$ | $0.001$ |
|---------------------------|---------|---------|---------|---------|---------|
| $k_{0}$                   | $2.706$ | $3.841$ | $5.024$ | $6.635$ | $10.828$ |

Double-blind experiment: In the experimental process, both the tester and the subject do not know which group the subject belongs to (experimental group or control group). When analyzing the data, the analyst usually does not know which group the data being analyzed belongs to, aiming to eliminate possible subjective biases and personal preferences that may exist in the tester and participant's consciousness.  
Placebo: It refers to a substance that has no therapeutic effect, but looks similar to the actual medication in terms of appearance such as tablets, pills, or injections.

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