
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 student interest group randomly surveyed the air quality level of each day in a city for 100 days and the number of people exercising in a park on that day. The data is summarized in the table below (unit: days):

| Exercise Times | Air Quality Level |
|----------------|-------------------|
| $[0,200]$      | $(200,400]$       | $(400,600]$       |
| $1 (Excellent)$| $2$               | $16$              | $25$              |
| $2 (Good)$     | $5$               | $10$              | $12$              |
| $3 (Mild Pollution)$ | $6$          | $7$               | $8$               |
| $4 (Moderate Pollution)$ | $7$     | $2$               | $0$               |

$(1)$ Estimate the probabilities of the air quality levels being $1$, $2$, $3$, and $4$ in the city for a day;  
$(2)$ Find the estimated average number of people exercising in the park in a day (representing the data in the group by the midpoint value of the interval);  
$(3)$ If the air quality level of a day is $1$ or $2$, it is considered "good air quality"; if the air quality level of a day is $3$ or $4$, it is considered "poor air quality." Complete the $2\times 2$ contingency table below based on the given data, and based on the contingency table, determine whether there is a $95\%$ confidence that the number of people exercising in the park on a day is related to the air quality of the city on that day.

|                 | Exercise Times $\leq 400$ | Exercise Times $> 400$ |
|-----------------|---------------------------|------------------------|
| Good Air Quality|                           |                        |
| Poor Air Quality|                           |                        |

Given: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$

| $P(K^{2}\geq k)$ | $0.050$ | $0.010$ | $0.001$ |
|------------------|---------|---------|---------|
| $k$              | $3.841$ | $6.635$ | $10.828$ |

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