Count Beautiful Splits in an Array

For the known beautiful-splits definition, enumerate two cut points and reduce each prefix-equality condition to a longest-common-prefix query between array suffixes. Build an LCP table from right to left: if nums[i] equals nums[j], lcp[i][j] = 1 + lcp[i+1][j+1], otherwise zero. Then a segment beginning at a is a prefix of one beginning at b when lcp[a][b] is at least the required segment length. Check both allowed relationships from the specification, while ensuring all three parts are nonempty and lengths are valid. This avoids comparing subarrays repeatedly inside the double loop. Be explicit about the exact split condition before coding, as it is easy to reverse an index. Typical preprocessing is O(n squared) time and space, followed by O(n squared) split enumeration.