
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 certain school has two restaurants, $A$ and $B$, to investigate the students' satisfaction with the restaurants. During a meal, the school randomly selected $67$ students from restaurant $A$ and $69$ students from restaurant $B$. Among them, $15$ students from restaurant $A$ and $6$ students from restaurant $B$ were dissatisfied with the service, while the rest were satisfied.  
$(1)$ Based on the data, construct a $2\times 2$ contingency table. Using the significance level $\alpha = 0.005$, can it be concluded that there is an association between the students' satisfaction and the two restaurants?  
$(2)$ The school conducted a statistical analysis on a large number of students who dined. They concluded the following: Any student, on their first visit to the school cafeteria, equally likely chooses one of the two restaurants. Starting from the second visit, if they went to restaurant $A$ on the previous visit, the probabilities of going to restaurants $A$ and $B$ on the current visit are $\frac{1}{4}$ and $\frac{3}{4}$, respectively. If they went to restaurant $B$ on the previous visit, the probabilities of going to restaurants $A$ and $B$ on the current visit are both $\frac{1}{2}$. Find the probability that any student will visit restaurant $B$ on their third visit.  
Given: ${\chi^2} = \frac{{n{{(ad-bc)}^2}}}{{(a+b)(c+d)(a+c)(b+d)}}$, where $n=a+b+c+d$.  
| $\alpha $ | $0.100$ | $0.050$ | $0.025$ | $0.010$ | $0.005$ |
|-----------|---------|---------|---------|---------|---------|
| $x_{0}$   | $2.706$ | $3.841$ | $5.024$ | $6.635$ | $7.879$ |

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