
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
To popularize knowledge of fire safety, a certain school organized a competition on related knowledge. The competition is divided into two rounds, and each participant must participate in both rounds. If a participant wins in both rounds, they are considered to have won the competition. It is known that in the first round of the competition, the probabilities of participants A and B winning are $\frac{4}{5}$ and $\frac{3}{5}$, respectively; in the second round, the probabilities of A and B winning are $\frac{2}{3}$ and $\frac{3}{4}$, respectively. A and B's outcomes in each round are independent of each other.<br/>$(1)$ The probability of A winning exactly one round in the competition;<br/>$(2)$ If both A and B participate in the competition, find the probability that at least one of them wins the competition.

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