
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 independently decode a password. The probabilities that they can decode the password are $\frac{1}{3}$ and $\frac{1}{4}$, respectively. Find:<br/>
$(1)$ The probability that both of them can decode the password;<br/>
$(2)$ The probability that neither of them can decode the password;<br/>
$(3)$ The probability that exactly one of them can decode the password.

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