
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
The influenza weekly report for the 8th week of 2023 released by the Chinese National Influenza Center on March 2nd stated that the positivity rate of influenza virus detection in southern and northern provinces continued to rise. A hospital uses two treatments, A and B, to treat influenza patients. In order to understand the effectiveness of the two treatment plans, 105 patients were randomly selected for investigation of their recovery period. The contingency table is as follows (Note: a recovery period longer than 7 days is considered a long recovery period):

| Treatment/Number | Long Recovery | Short Recovery |
|------------------|----------------|----------------|
| A                | 10             | 45             |
| B                | 20             | 30             |

$(1)$ Is there a $95\%$ confidence that the "recovery period" is related to the treatment plan?  
$(2)$ Now, using stratified random sampling, 10 people are randomly selected from the sample using treatment B, and then 3 people are randomly selected from these 10 people. Find the distribution table and expectation of the number $X$ of people with a long recovery period among the 3 selected.  
$(3)$ Assuming the recovery period for treatment A is denoted by $Y$, it is found that $Y$ approximately follows a normal distribution $N(5,1)$. If a patient is treated with plan A, is there more than a $95\%$ chance of recovery after 7 days? Explain the reason.

Given: $\chi ^{2}=\frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$  
| $P(\chi ^{2}\geqslant x_{0})$ | 0.1   | 0.05  | 0.010 |
|-----------------------------|-------|-------|-------|
| $x_{0}$                     | 2.706 | 3.841 | 6.635 |

If $\xi \sim N(\mu, \sigma ^{2})$, then $P\left(\mu -\sigma\ \ \lt \xi\ \ \lt \mu +\sigma \right)=0.6862$, $P\left(\mu -2\sigma\ \ \lt \xi\ \ \lt \mu +2\sigma \right)=0.9544$, $P\left(\mu -3\sigma\ \ \lt \xi\ \ \lt \mu +3\sigma \right)=0.9974$.

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