
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
Handan is a city with a rich historical and cultural heritage, known as the "capital of Chinese idioms and allusions." In order to help the citizens better understand and inherit the culture of idioms, the local cultural and tourism bureau plans to hold a competition to guess idioms. There are a total of $n$ questions in the competition. Participants start answering from the first question, and once they answer incorrectly, they stop answering. Otherwise, they continue until all questions are answered. Let a player's probability of answering each question correctly be $p\left(0 \lt p \lt 1\right)$, and the correctness of each question is independent of others.  
$(1)$ Let $X$ denote the number of questions answered when the answering ends. When $n=3$, if $E\left(X\right) \gt 1.75$, find the range of values for $p$;  
$(2)$  
$(i)$ Let $Y$ denote the number of questions answered correctly when the answering ends. Find $E\left(Y\right)$;  
$(ii)$ When $p=\frac{5}{6}$, find the minimum value of $n$ that makes $E\left(Y\right) \gt 4$.  
Reference data: $\lg 2\approx 0.301$, $\lg 3\approx 0.477$.

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