Here we´ll provide more detailed instructions on how to make use of Hint 2. To begin with, we introduce the following reference frames and 4-velocities:
Frame (A), the laboratory frame, and its 4-velocity u;
Frame (B), the spaceship after the first acceleration stage, and its 4-velocity v;
Frame (C), the spaceship after the second acceleration stage, and its 4-velocity w.
As a first step, we derive the 4-components of u in frame (C), and the components of w in frame (A). To that end, we make use of the equalities relating the expressions of the 4-invariants in one frame to the their expressions in another frame, e.g. u^A_tv^A_t-u^A_xv^A_x-u^A_yv^A_y=u^B_tv^B_t-u^B_xv^B_x-u^B_yv^B_y (we have dropped the z-components as we assume motion in the x-y-plane). As a second step, we introduce one more frame;
Frame (D), the spaceship after the fourth acceleration stage, and its 4-velocity q. Note that the expression we derived during the first step can also be used to express the components of q in the frame (C) (be careful, it is easy to make a mistake here). The 4-invariants relating q and u can be used to obtain a closed equation for the speed V of the spaceship after the first acceleration stage.
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