
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]

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[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

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The Hangzhou Asian Games are scheduled to be held from September 23 to October 8, 2023. During this period, athletes participating in the Asian Games can stay and eat for free at the Asian Games Village. A certain restaurant in the Asian Games Village offers only "Chinese set meal" and "Western set meal" for dinner from the first day to the last day. It is known that a certain athlete will choose between these two set meals for dinner every day. It is given that the probability of choosing the "Chinese set meal" on the first night is $\frac{4}{5}$, the probability of continuing to choose the "Chinese set meal" the next night after choosing it the previous night is $\frac{1}{4}$, and the probability of continuing to choose the "Western set meal" the next night after choosing it the previous night is $\frac{1}{3$, and so on.

$(1)$ Find the probability that the athlete chooses the "Chinese set meal" on the second night.
$(2)$ Let the probability that the athlete chooses the "Chinese set meal" on the $n$th night $(n=1,2,\ldots,16)$ be $P_{n}$.
$(i)$ Find $P_{n}$.
$(ii)$ Find the number of nights out of the 16 nights where the probability of choosing the "Chinese set meal" is greater than the probability of choosing the "Western set meal".

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