Simulate robot path and detect boundedness
Overview
This question evaluates the ability to simulate iterative movement and reason about repeated state transformations, testing competencies in algorithm design, cycle detection, and geometric reasoning; it falls under the Coding & Algorithms category.
A robot starts at (0, 0) facing north on an infinite 2D plane. It executes a command string s consisting of 'G' (move forward one unit), 'L' (turn left 90°), and 'R' (turn right 90°). After finishing s once, the robot repeats the same string k times, where k can be as large as 10^9. Implement an algorithm that (a) returns the robot’s final position and facing direction after one execution of s, and (b) determines whether the trajectory remains within a bounded region if s is repeated indefinitely. Provide time and space complexity and avoid simulating all k repetitions.