public static List<String> findMissingRanges(int[] nums, int lower, int upper) {
        List<String> missingRanges = new ArrayList<>();
        long next = lower; // use long to avoid overflow issues

        for (int num : nums) {
            if (num > next) {
                missingRanges.add(formatRange(next, num - 1));
            }
            next = (long) num + 1; // move to next expected number
        }

        if (next <= upper) {
            missingRanges.add(formatRange(next, upper));
        }

        return missingRanges;
    }

    private static String formatRange(long start, long end) {
        return start == end ? String.valueOf(start) : start + "->" + end;
    }