
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 attempt to decrypt a password. Their probabilities of successfully decrypting the password are $\dfrac{1}{3}$ and $\dfrac{1}{4}$, respectively. Calculate:

$(1)$ The probability that exactly one of them decrypts the password.

$(2)$ If the probability of decrypting the password needs to be $\dfrac{99}{100}$, what is the minimum number of people like B required?

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