
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 held a knowledge competition to strengthen campus safety and enhance students' safety awareness. Teams of students participated in the competition, with each team earning $5$ points for each correct answer and $1$ point for each incorrect answer. It is known that the correct rates of three types of related knowledge questions in the question bank for Team A are as follows:

| Category     | Traffic Safety | Fire Safety | Water Safety |
|--------------|----------------|-------------|--------------|
| Correct Rate | $\frac{2}{3}$  | $\frac{1}{2}$| $\frac{1}{3}$ |

$(1)$ If Team A draws one question each from Traffic Safety and Fire Safety, find the probability that Team A will score at least $6$ points after answering these two questions;  
$(2)$ It is known that Team A draws $3$ questions, and the categories are all different. Let $Y$ be the total score of Team A after answering these $3$ questions. Find the distribution and mathematical expectation of $Y$.

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