
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
Person A and person B each shoot once, with probabilities of hitting the target being $\frac{2}{3}$ and $\frac{3}{4}$, respectively. Assume that each shot is independent of the others. Find the following probabilities in terms of fractions:
- $(1)$ The probability that person A misses the target at least once in 3 shots.
- $(2)$ The probability that person A hits the target exactly twice and person B hits the target exactly once in 2 shots each.

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