Write a program that finds the longest word in a string and outputs it along with the index of its first occurrence.

1. The program should split the string into words and check the length of each word.
2. For each word, compare its length with the length of the current longest word.
3. If the current word is longer than the longest one, update the longest word and its index.
4. Finally, output the result in the format: "Longest word: [word] at index [index]", where [word] is the longest word and [index] is the index of its first occurrence in the string.