[BUG] The action fails to authenticate to private npm registry

### Action config

This is a slightly modified (from default generated) workflow that I have:

```yml
# This file was auto-generated by the Firebase CLI
# https://github.com/firebase/firebase-tools

name: Deploy to Firebase Hosting on merge
on:
push:
branches:
- develop
jobs:
build_and_deploy:
runs-on: ubuntu-latest
steps:
- uses: actions/checkout@v4
- name: Set up Node.js
uses: actions/setup-node@v4
with:
node-version: 18
registry-url: "https://npm.pkg.github.com"
scope: "@org-name"
always-auth: true
- run: npm ci && npm run build
env:
NODE_AUTH_TOKEN: ${{ secrets.NPM_TOKEN }}
- uses: FirebaseExtended/action-hosting-deploy@v0
with:
repoToken: ${{ secrets.GITHUB_TOKEN }}
firebaseServiceAccount: ${{ secrets.FIREBASE_SERVICE_ACCOUNT_FIREBASE_SANDBOX_425310 }}
channelId: live
projectId: firebase-sandbox-425310
env:
FIREBASE_CLI_EXPERIMENTS: webframeworks,pintags
NODE_AUTH_TOKEN: ${{ secrets.NPM_TOKEN }}
```

### Error message

Firebase deployment fails with `nnpm ERR! 401 Unauthorized - GET https://npm.pkg.github.com/download/@org-name/app-components-frontend/0.18.33/233b5141a9f47e0a5863ebb4cd714b82c4c87100 - authentication token not provided`

### Expected behavior

While `npm ci` works fine because it uses an `.npmrc` file generated by the `setup-node` action, firebase hosting deploy action doesn't seem to be picking that up.

I've tried to find something in the docs of either firebase tools or the action, but didn't find anything that would help me resolve this.

### Actual behavior

Auth to npm registry fails and hence deployment fails

Start by reproducing the workflow using actions/setup-node, the generated .npmrc, and NODE_AUTH_TOKEN, then inspect how FirebaseExtended/action-hosting-deploy runs the deployment. Compare the successful npm ci step with the failing Firebase deployment and verify that a private npm package can authenticate during deployment without exposing the token.

Written by the indexing model from the issue text.