The root finder must return a result backed by a valid sign-changing interval. Finish the solver so endpoint roots and the stopping rule are handled without silently losing the bracket.
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 nonempty finite coefficients in descending power order, finite lo <= hi and positive finite tolerance. If either endpoint evaluates exactly to zero, return it. Otherwise signs must differ. Repeatedly bisect, return an exact midpoint root immediately, otherwise replace the endpoint with the same sign. Stop when bracket width <= tolerance and return midpoint rounded to six decimals. Reject invalid brackets or parameters. Inputs must converge within 200 bisections.
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.