
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
Groups \\(A\\) and \\(B\\) each have \\(7\\) patients, and their recovery times (in days) after taking a certain medication are recorded as follows:  
Group \\(A\\): \\(10\\), \\(11\\), \\(12\\), \\(13\\), \\(14\\), \\(15\\), \\(16\\)  
Group \\(B\\): \\(12\\), \\(13\\), \\(15\\), \\(16\\), \\(17\\), \\(14\\), \\(a\\)  
Assuming the recovery times of all patients are independent, one person is randomly selected from each of groups \\(A\\) and \\(B\\), with the person from group \\(A\\) denoted as \\(\alpha\\) and the person from group \\(B\\) denoted as \\(\beta\\).  
\\((\\)I\\()\\) Calculate the probability that \\(\alpha\\)'s recovery time is at least \\(14\\) days;  
\\((\\)II\\()\\) If \\(a=25\\), calculate the probability that \\(\alpha\\)'s recovery time is longer than \\(\beta\\)'s recovery time;  
\\((\\)III\\()\\) For what value of \\(a\\) will the variances of the recovery times for groups \\(A\\) and \\(B\\) be equal? \\((\\)No proof required for the conclusion\\()\\)

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