The trajectory exporter needs explicit Euler updates for the supplied linear system. Implement the integration so every state in a step is computed from the preceding state.
Build the implementation in C++ under /app, with /app/run.sh as its entry point. Read one JSON request from standard input and write one JSON result to standard output. Input has finite numbers initial, a, b, positive dt and nonnegative integer steps. Starting at y=initial, update y += dt*(a*y+b) exactly steps times. Return the initial and every resulting y rounded to six decimals, without rounding internal updates. Reject invalid fields, boolean steps or nonpositive dt. Inputs are bounded so all iterates remain finite.
Reject inputs that violate an explicit validation rule with exit status 2, no standard output, and {"error":"invalid_input"} on standard error. For valid input, exit with status 0. Keep the implementation reusable for new inputs; it must work offline and must not depend on the example data being the only data it receives.
Repair or replace the unfinished /app/run.sh and add source files under /app. Malformed JSON must use the same invalid-input response. Do not read /tests or /oracle from the application.