. . . . . . . . few days later I entered . . . . . . . . reading room of . . . . . . . . public library.

In the first blank, article "a" will be used because article "a" is used when the speaker is mentioning something for the first time or the speaker is making a general statement about any such thing.
In the second blank, article "the" will be used because article "the" is used to refer to something specific. Here, it is written that the room is specifically the 'reading room'.
Similarly, in the third blank, it is specifically mentioned that the library is 'public'. Therefore, article "the" will be used.