
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
Jinzhong City is the birthplace of Jin merchants' culture and has rich tourism resources, including well-preserved cultural landscapes (such as Wang's Courtyard, Chang's Manor, etc.) and beautiful natural landscapes (such as Mianshan in Jiexiu, Gypsum Hill, etc.). A tour group brings tourists to Jinzhong for sightseeing, where tourists can freely choose to visit either a cultural or a natural landscape. If the probability of each tourist choosing a cultural landscape is $\frac{2}{3}$ and choosing a natural landscape is $\frac{1}{3}$, and the willingness of tourists to choose is independent of each other.<br/>$(1)$ Randomly select $5$ people from the tourists, let $X$ be the number of people who choose a cultural landscape among the $5$ people, find the mean and variance of $X$;<br/>$(2)$ Now, a survey is conducted on the tourists. If choosing a cultural landscape earns $2$ points and choosing a natural landscape earns $1$ point, the probability of the cumulative score being $n$ points after the survey is $P_{n}$, find $P_{n}$.<br/>

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