The number of nouns after another needs to be determined on a case-by-case basis. In general, if it refers to "the other three", add the plural form of the noun, such as "the other three apples", and use "another three apples". If it refers to "another", add the singular form of the noun, such as "another apple", and use "another apple".
When the singular number of countable nouns is added after another, it means that it refers to the "third" in the same class of people or things, that is, "another", not "three". Thus, another can be used in conjunction with the singular number of countable nouns to mean "another". For example: another day, another man.
When another is followed by a plural number of countable nouns, it means that it refers to "several others" in the same type of person or thing, that is, "some others". Thus, another can be used in conjunction with the plural of countable nouns to mean "several others". For example: another three days, another three books.
It should be noted that when an uncountable noun is added after another, it means that it refers to "another" in the same type of person or thing, that is, "another uncountable noun". Therefore, another cannot be used in conjunction with an uncountable noun. For example: another word, another piece of news.
To sum up, the number of nouns after another needs to be determined on a case-by-case basis. If it refers to "the other three", add the noun plural; If it refers to "another", add the singular form of the noun; When an uncountable noun is added after another, it means that it refers to "another" in the same class of people or things, that is, "another uncountable noun".