
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
To enhance students' humanistic literacy and cultivate their interest in literary studies, a certain school held a poetry competition. The competition consists of $3$ required questions and $3$ buzzer questions. Both sides must provide answers to the required questions, scoring $1$ point for a correct answer and no points for a wrong answer; the buzzer questions are answered by the side that buzzes in first, scoring $2$ points for a correct answer and losing $1$ point for a wrong answer. After both segments, the side with the higher total score wins. Due to students' general feedback that the fairness of this format is insufficient, the school will reform the competition format: adjusting to $4$ required questions and $2$ buzzer questions, with the point values for each question remaining the same.

$(1)$ To test the impact of the new format on participants' performance, the school selected two students, A and B, to practice under both formats and recorded the outcomes of their matches. Please complete the following $2\times 2$ contingency table based on the given information, and determine whether there is a $95\%$ confidence that the winner is related to the format.

\[
\begin{array}{|c|c|c|c|}
\hline
& \text{Old Format} & \text{New Format} & \text{Total} \\
\hline
\text{A Wins} & 6 & 9 & 15 \\
\hline
\text{B Wins} & 4 & 1 & 5 \\
\hline
\text{Total} & 10 & 10 & 20 \\
\hline
\end{array}
\]

$(2)$ Student C is good at buzzer questions. It is known that the probability of C buzzing in to answer a buzzer question is $0.6$, the probability of answering a buzzer question correctly is $0.8$, and the probability of answering a required question correctly is $p\left(0 \lt p \lt 1\right)$. The responses to each question are independent.

$($i$)$ Let $X$ denote the score C receives for a buzzer question. Find the probability distribution and the expected value of $X$.

$($ii$)$ Given that the absolute difference between the expected total scores of student C under the new and old formats does not exceed $0.1$ point, find the range of values for $p$.

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

\[
\begin{array}{|c|c|c|c|c|}
\hline
P(K^{2}\geqslant k_{0}) & 0.15 & 0.10 & 0.05 & 0.025 \\
\hline
k_{0} & 2.072 & 2.706 & 3.841 & 5.024 \\
\hline
\end{array}
\]

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