
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 holds a math competition, where participants need to complete three questions: algebra, geometry, and combinatorics. The scoring method for the competition is as follows: within the specified time, answering the algebra and geometry questions correctly earns $30$ points each, while answering the combinatorics question correctly earns $40$ points. For each incorrect answer, $10$ points are deducted from the total score (assuming there are only two possible outcomes for each question: correct or incorrect). Based on past statistical results, the probabilities of Xiao Ming answering the algebra, geometry, and combinatorics questions correctly are $\frac{3}{4}$, $\frac{2}{3}$, and $a$, respectively. Assuming the results of answering these three questions are independent of each other, and given that Xiao Ming starts with a score of $0$ points, let $X$ be Xiao Ming's total score after the competition. The questions are as follows:

$(1)$ Given that the probability of Xiao Ming answering all three questions correctly within the specified time is $\frac{1}{6}$, find the probability that the student can answer exactly one of the three questions correctly.

$(2)$ Given that $a=\frac{1}{2}$, and Xiao Ming needs a score of at least $50$ points to win a prize, find the probability of Xiao Ming winning a prize.

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