
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 hosting of the Beijing Winter Olympics has not only attracted 300 million people to participate in ice and snow sports but also ignited the enthusiasm for national fitness. The development of ice and snow sports and the smooth advancement of national fitness have laid a solid foundation for building a sports powerhouse. With the implementation of the ice and snow sports strategy of "expanding to the south, west, and east," ice and snow sports are no longer limited to some traditional ice and snow provinces and cities. In order to understand the participation of citizens in ice and snow sports, a certain survey center randomly selected 100 people from City $A$ and City $B$ each to investigate whether these people have participated in ice and snow sports. The $2\times 2$ contingency table obtained is as follows:

|           | Participated in ice and snow sports | Did not participate in ice and snow sports | Total |
|-----------|-------------------------------------|--------------------------------------------|-------|
| City $A$  | 60                                  | 40                                         | 100   |
| City $B$  | 30                                  | 70                                         | 100   |
| Total     | 90                                  | 110                                        | 200   |

$(1)$ Complete the $2\times 2$ contingency table and determine whether there is a 99.9% certainty that participation in ice and snow sports is related to the city.

$(2)$ Based on the statistical table, using stratified sampling by city, select 6 people from those who have participated in ice and snow sports, then randomly select 2 people from these 6 people. Find the probability of exactly 1 person from City $A$ and 1 person from City $B$ being selected.

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

| $P(K^{2}\geqslant k_{0})$ | 0.100 | 0.050 | 0.025 | 0.010 | 0.001 |
|---------------------------|-------|-------|-------|-------|-------|
| $k_{0}$                   | 2.706 | 3.841 | 5.024 | 6.635 | 10.828|

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