As the fourth hint, let us write a few starting lines of the solution.
The 4-velocity u of the frame (A) has the components (1,0,0) in frame (A), the components (\gamma c, -\gamma V, 0) in frame (B), and the components (\gamma' c, -\gamma' P, \gamma' Q) in frame (C). Here V denotes the speed of the spaceship after the first stage of acceleration (which we want to determine), P and Q are the unknown components of the velocity of the laboratory frame in the frame (C), \gamma=1/\sqrt{1-V^2/c^2}, and \gamma'=1/\sqrt{1-(P^2+Q^2)/c^2}. Next, the 4-velocity v of the frame (B) in frames (A), (B), and (C) is written as (\gamma c, \gamma V, 0), (1,0,0), and (\gamma c, 0, -\gamma V), respectively. It should now be clear how to proceed, and use the instructions in Hint 3.
Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The next hint for the Problem 4 will be published at 13:00 GMT, 26th March 2023, together with the next updates of the intermediate results.