
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
A certain school organized a "My Country is Awesome" knowledge competition, consisting of 25 multiple-choice questions, each with 4 options, of which only 1 is correct. The test has a total score of 150 points, with 6 points awarded for each correct answer, 2 points for not answering, and 0 points for a wrong answer. Candidates A and B have both answered the first 20 questions correctly. They are unsure about the last 5 questions (denoted as $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$, $T_{5}$). Both of them can choose to answer some of these 5 questions. When answering, if they can eliminate some wrong options, they will randomly choose 1 from the remaining options; otherwise, they will randomly choose 1 from all 4 options. It is known that A can only eliminate 1 wrong option from $T_{1}$, $T_{2}$, $T_{3}$, so A decides to answer only these three questions and give up $T_{4}$ and $T_{5}$.

$(1)$ Find the probability that A's total score is not less than 135 points.

$(2)$ Find the probability distribution of A's total score.

$(3)$ Given that B can eliminate 2 wrong options from $T_{1}$, $T_{2}$, $T_{3}$, 1 wrong option from $T_{4}$, but cannot eliminate any wrong options from $T_{5}$. What answering strategy should B adopt for the last 5 questions to maximize the expected total score, and explain the reason.

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