
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
Multiple-choice questions are a type of question in the new college entrance examination. The rules are as follows: if all the selected options are correct, the student gets $5$ points; if some are correct, the student gets $2$ points; if any are incorrect or if none are selected, the student gets $0$ points. A student is taking the midterm exam in Xichang City. When he reaches questions $11$ and $12$, he realizes he doesn't know the answers. In these two multiple-choice questions, the probability of him selecting one option is $\frac{1}{3}$, selecting two options is $\frac{1}{3}$, and selecting three options is $\frac{1}{3}$. If the student guesses the answers independently for each question and the correct answers for both questions $11$ and $12$ are two options.<br/>$(1)$ Find the probability that the student gets $2$ points for question $11$;<br/>$(2)$ Find the probability that the student scores a total of $7$ points for questions $11$ and $12$.

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