
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
Recently, various departments in Jiujiang City have launched a wave of creating civilized cities. In order to enhance the awareness of creating a national civilized city among teachers and students, a certain school organized a knowledge assessment for teachers to create a national civilized city. Each teacher must participate and can participate in the assessment at most 2 times. Once the first assessment is passed, the teacher will not participate in the second assessment. Teachers who do not pass the assessment twice will have their civilization points deducted. It is known that the probability of Teacher A passing each assessment is $\frac{2}{3}$, and the probability of Teacher B passing each assessment is $\frac{1}{2}$, and whether Teacher A and Teacher B pass each assessment is independent of each other.  
$(1)$ Find the probability of Teacher A passing the assessment;  
$(2)$ Find the probability that the total number of assessments for Teacher A and Teacher B is 3.

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