
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 city randomly selects monitoring data of the Air Quality Index (API) for 100 days within a year of 365 days. The statistics are as follows:

| API       | $[0,50]$ | $(50,100]$ | $(100,150]$ | $(150,200]$ | $(200,250]$ | $(250,300]$ | $> 300$ |
|-----------|----------|------------|-------------|-------------|-------------|-------------|---------|
| Air Quality | Excellent | Good | Slight Pollution | Mild Pollution | Moderate Pollution | Moderate Severe Pollution | Severe Pollution |
| Days      | 4        | 13         | 18          | 30          | 9           | 11          | 15      |

$(1)$ If the economic loss $S$ (unit: yuan) caused by air pollution for a company each day is related to the Air Quality Index (denoted as $\omega$) by the formula: $S=\left\{\begin{array}{l}0,0\leq \omega \leq 100\\ 4\omega-400,100<\omega \leq 300\\ 2000,\omega > 300\end{array}\right.$, estimate the probability that on a randomly selected day in the year, the economic loss $S$ exceeds 200 yuan but does not exceed 600 yuan;<br/>$(2)$ If 30 days of the sample data are in the heating season, with 8 days being severely polluted, complete the following $2\times 2$ contingency table, and determine whether there is a 95% confidence that the severe air pollution in the city this year is related to heating:

|              | Non-Severe Pollution | Severe Pollution | Total |
|--------------|----------------------|------------------|-------|
| Heating Season | 22                 | 8                | 30    |
| Non-Heating Season | 63             | 7                | 70    |
| Total        | 85                 | 15               | 100   |

Given: $P(K^{2}\leq k_{0})$

| $P(K^{2}\leq k_{0})$ | 0.25 | 0.15 | 0.10 | 0.05 | 0.025 | 0.010 | 0.005 | 0.001 |
|----------------------|------|------|------|------|-------|-------|-------|-------|
| $k_{0}$             | 1.323| 2.072| 2.706| 3.841| 5.024 | 6.635 | 7.879 | 10.828|

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

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