Reach, gain, compass, achieve, attain can mean to arrive at a point by effort or work.
Reach is the most general term, being capable of reference to whatever can be arrived at by exertion of any degree and applicable to such diverse matters as a point in space, in time, or in a development, or as a destination, a goal, or a position of eminence.
In extended use reach may be predicated even of inanimate things.
Gain usually implies a struggle to reach a contemplated or desired destination or goal.
Compass implies efforts to get around difficulties or to transcend limitations; it often connotes skill or craft in management.
Achieve can stress the skill or the endurance as well as the efforts involved in reaching an end. Often it implies accomplishment of something that is in itself a feat or triumph.
Attain connotes more strongly than any of the others the spur of aspiration or ambition. It is therefore especially referable to ends beyond the vision, the scope, or the powers of most men.