Other and another: when do we use each one of them?

what is the best way to differentiate between other and another?
when do we use each one of them?
is another a combination of both an + other

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Yes.

‘Other’ is an adjective only; ‘another’ is both an adjective and a pronoun.
‘Other’ as an adjective must either apply to a plural noun (I have other friends) or be preceded by a determiner, possessive, etc (The/This/Her other dog is my friend).
‘Another’ as an adjective is just one form of ‘other’-- ‘other’ + indefinite article.
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