The numerical helper needs both a polynomial's value and its derivative at a requested point. Finish the evaluation so coefficient order and constant polynomials are handled correctly.
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 coefficients in descending power order and points, all finite numbers. At each point return value and derivative rounded to six decimals. Use Horner recurrence; an empty coefficient list denotes the zero polynomial. Reject boolean, nonnumeric or nonfinite inputs. Inputs are bounded so intermediate results 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.