ABC Conjecture 3

You are given a string 𝑆 of length 𝑁, containing only the characters a, b, and c.

In one move, you can modify 𝑆 as follows:

• Choose three indices 𝑖,𝑗,𝑘 (1≤𝑖<𝑗<𝑘≤∣𝑆∣) such that 𝑆𝑖=a,𝑆𝑗=b,𝑆𝑘=c
• That is, choose some subsequence of 𝑆 that equals abc.
• Then, delete either the a or the c from 𝑆, that is, either index 𝑖 or index 𝑘.
• Note that this reduces the length of 𝑆 by 1.

For example, if 𝑆=cbba‾cb‾ac‾c, you can choose 𝑖=4,𝑗=6,𝑘=8 (the underlined indices), and then delete either index 4 (obtaining 𝑆=cbbcbacc) or index 8 (obtaining 𝑆=cbbacbac).

Find the minimum number of moves that can be made on 𝑆, such that it's impossible to perform any further moves on the resulting string.