
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]

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[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

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Vaccination is one of the important means to protect one's own and others' health and lives. In order to test the immune effect of a certain vaccine on the $C$ virus, researchers used white rabbits as experimental subjects and conducted the following experiments:<br/>Experiment 1: Select 10 healthy white rabbits, numbered 1 to 10, inject them with the vaccine once, and then expose them to an environment containing the $C$ virus. The results of the experiment showed that four of the white rabbits, numbers 2, 6, 7, and 10, were still infected with the $C$ virus, while the others were not infected.<br/>Experiment 2: The vaccine can be injected a second time, but the time interval between the two injections must be more than three weeks. After injecting the vaccine, if antibodies can be produced in the organism, it is considered effective, and the vaccine's protection period is eight months. Researchers injected the first dose of the vaccine to the white rabbits three weeks after the first dose, and whether the vaccine is effective for the white rabbits is not affected by each other.<br/>$(1)$ In "Experiment 1," white rabbits numbered 1 to 5 form Experiment 1 group, and white rabbits numbered 6 to 10 form Experiment 2 group. Researchers first randomly select one group from the two experimental groups, and then randomly select a white rabbit from the selected group for research. Only one white rabbit is taken out each time, and it is not put back. Find the probability that the white rabbit taken out in the first time is infected given that the white rabbit taken out in the second time is not infected;<br/>$(2)$ If the frequency of white rabbits not infected with the $C$ virus in "Experiment 1" is considered as the effectiveness rate of the vaccine, can the effectiveness rate of a white rabbit after two injections of the vaccine reach 96%? If yes, please explain the reason; if not, what is the minimum effectiveness rate of each vaccine injection needed to ensure that the effectiveness rate after two injections of the vaccine is not less than 96%.

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