
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
A certain selection process consists of three rounds of assessments, with one question in each round. Those who answer the question correctly move on to the next round, otherwise they are eliminated. It is known that player A has probabilities of $\frac{4}{5}, \frac{3}{4}, \frac{2}{3}$ of correctly answering the questions in the first, second, and third rounds, respectively, while player B has probabilities of $\frac{2}{3}, \frac{2}{3}, \frac{1}{2}$ for the three rounds, respectively. The ability of the two players to answer questions correctly in each round is independent of each other.<br/>$(1)$ Find the probability that player A is eliminated only after entering the third round;<br/>$(2)$ Find the probability that at least one player passes all assessments.

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