
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
Two people, A and B, each shoot once. The probability of hitting the target for A and B are $\frac{2}{3}$ and $\frac{3}{4}$, respectively. It is assumed that each shot is independent of the others. Find the probabilities in the following scenarios, expressing the results as fractions:
- $(1)$ The probability that A shoots 3 times and misses at least once.
- $(2)$ The probability that A hits the target exactly 2 times and B hits the target exactly 1 time after each shooting twice.
- $(3)$ The probability that A misses the target and B hits the target 2 times after each shooting twice.
- $(4)$ The probability that both A and B hit the target once after each shooting twice.

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