Merging this claude-code-base-action into this repo

[ashwin-ant]

We'll be merging the code in https://github.com/anthropics/claude-code-base-action/ into this repo fairly soon. The reason is primarily around developer ergonomics. We'll still keep the base action repo around, likely as a mirror of the base action code in this repo. This will simplify a lot of refactoring that we're hoping to do soon, and should hopefully have no impact on end users.