Z-algorithm matching count

For a string s, consider each starting index i (including 0). Index i counts if the substring of s that begins at i shares at least one leading character with s itself — equivalently, the Z-value z[i] is positive (and z[0] is the whole string by convention). Return how many indices count. For example, in "aabaab" the qualifying indices are 0, 1, 3, 4, so the answer is 4. The string can be up to 10^5 characters; aim for linear time.