The measurement report needs a weighted center and radial spread. Implement the calculation so weights influence both results consistently and invalid inputs follow the stated rules.
Build the implementation in Python 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 is a nonempty list of points with x, y and weight, all finite numeric values. Weights must be nonnegative and total weight positive. Return centroid [weighted x, weighted y] and radius2, the weighted mean squared distance to that centroid, rounded to six decimals. Reject invalid weights or nonfinite values.
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.