Function apply_patch_and_push 

pub async fn apply_patch_and_push(
    patch_path: &Path,
    repo_root: &Path,
    branch: Option<&str>,
    base: &str,
    title: &str,
    dco_signoff: bool,
    force: bool,
    progress: impl Fn(PatchStep),
) -> Result<String, PatchError>

Apply a patch file, commit, and push to origin. Returns the branch name that was pushed.