Read the following sentences describing the classification problem and answer the questions.

Given data consisting of variable $$

and variable $1,...,$

$,$ consider classifying the variable $$

with the objective of determining the regions that fit the data well. No data generation and observation mechanism is assumed.

If you consider that the distance between data belonging to different classes should be large and the distance between data belonging to the same class should be small, you may choose $[(1)]$ and $[(2)]$ as the criterion of goodness of fit, and determine the function that indicates the boundary of the region so that the ratio $[(1)]/[(2)]$ of the two is maximized. Such a classification method is known as $[(3)].$

Also, if you consider that the point closest to the boundary of the region should be as far away from the boundary as possible, then you can determine a function that indicates the region of the boundary so as to maximize $[(4)].$ Such a classification method is known as $[(5)].$