Minimum Insertions to Form Repeated abc Pattern

Given a lowercase string, insert the fewest characters needed so the result is a repetition of `abc` blocks. Deletion and replacement are not allowed. Requirements Input: a lowercase string s. Operation: insert any lowercase character at any position. Output: the minimum number of insertions needed to transform s into a string of the form "abcabcabc...". Original characters must remain in order; deletion and replacement are not allowed. Examples Input: "a" Output: 2 Insert b and c to form abc. Input: "aa" Output: 4 Insert b, c, b, and c to form abcabc. Notes Simulate matching against the infinite pattern abcabc.... Maintain one pointer into s and one pointer into the target pattern. When s[i] matches the expected pattern character, consume it; otherwise count an insertion. After all original characters are consumed, add the insertions needed to finish the current abc block. Preparation Solve the canonical abc valid-string problem three ways: direct expected-character simulation, block counting, and a small-state greedy scan. Rehearse tail handling separately; after consuming the input, the current partial block must still be completed.