Node distributions not pulled from authenticated mirrors

Description:
When you use an authenticated mirror, the node distribution is always pulled from github, not from the mirror. This is due to the manifest check always being run, and the mirror fallback runs only if the manifest fails or the version isn't found.

Action version:
v5 (but 6 appears to have the same problem)

Platform:

• Ubuntu
• macOS
• Windows

Runner type:

• Hosted
• Self-hosted

Tools version:
N/A

Repro steps:
Run with any authenticated mirror (in my case, jfrog artifactory). Set mirror and mirror-token, with node-version: 22.20.0 for example.

Expected behavior:
setup-node pulls the node distribution from my mirror https://redacted/artifactory/nodejs-remote/dist/v22.20.0/node-v22.20.0-linux-x64.tar.gz.

Actual behavior:
setup-node always pulls the node distribution from https://github.com/actions/node-versions/releases/download/22.20.0-17993202319/node-22.20.0-linux-x64.tar.gz.

I can provide a PR to solve this, if desired.

Start by tracing the setup-node handling for the mirror, mirror-token, and node-version inputs, focusing on the manifest check and mirror fallback described in the report. Reproduce with an authenticated Artifactory mirror and Node 22.20.0; done means the distribution is fetched from the mirror URL instead of GitHub.