Solution: Minimum Number of Pushes to Type Word II

Statement

The keys (numbered $2$ to $9$) on a telephone keypad are associated with specific groups of lowercase English letters. Below is an example of how letters can be mapped to keys on a telephone keypad.

Note: The keys $1$, $*$, $0$, and $\#$ do not map to any letters.

In the above mapping, the key $2$ is mapped with letters ["a", "b", "c"]; pressing it once types "a", pressing it twice types "b", and pressing it three times types "c".

Any mapping is possible as long as each letter is assigned to exactly one key, and keys can be mapped to any number of distinct letters.

Given a string, word, consisting of lowercase English letters, return the minimum number of pushes needed to type it after remapping ...