Firebase Functions Python build cannot reliably install private Artifact Registry packages

## Summary
Deploying Firebase Functions written in Python cannot reliably install private Python packages hosted in Google Artifact Registry using the documented authentication mechanism.

Although Artifact Registry supports authentication via the `keyrings.google-artifactregistry-auth` backend, the Firebase Functions build process installs dependencies in a single `pip` invocation. This creates a bootstrapping problem where the authentication backend must already be installed before pip attempts to resolve private dependencies.

As a result, private packages cannot be consumed out-of-the-box during function deployment causing deployment issues and ultimately meaning private packages cannot be consumed in firebase functions. Private Python packages are a common pattern for shared internal libraries, and the current limitation makes this workflow difficult.

## Expected Behaviour

A Firebase Functions deployment should be able to install private Python packages hosted in Artifact Registry without requiring custom build pipelines, vendored wheels, or containerization.

For example, the following requirements.txt should work:
```
--extra-index-url https://-python.pkg.dev///simple/

private-package==0.0.1
```

provided that:
- the Cloud Build service account has artifactregistry.reader
- the repository contains the package

## Actual Behaviour
The Artifact Registry authentication mechanism requires the keyring backend to be present before pip attempts to fetch packages.

However pip resolves and downloads packages before installing their dependencies, meaning that adding the backend as a dependency does not help.

Example attempt:
```
--extra-index-url https://-python.pkg.dev///simple/

keyring
keyrings.google-artifactregistry-auth
private-package==0.0.1
```

This still fails because pip attempts to resolve `private-package` before installing the keyring backend. This creates a bootstrap race condition

## Why this is a Problem

The current behavior makes it effectively impossible to use private Artifact Registry Python packages with Firebase Functions without workarounds such as:

- vendoring wheels into the repository
- migrating to cloud run where the build process can be controlled

## Environment Details

Builder image:
```
europe-west2-docker.pkg.dev/serverless-runtimes/google-22-full/builder/python:python_20260224_RC00_lightweight
```
Buildpacks used:
```
google.utils.archive-source
google.python.runtime
google.python.functions-framework
google.python.pip
google.utils.label-image
google.python.link-runtime
```
The pip step is executed as:
```
python3 -m pip install --requirement requirements.txt
```
Python runtime:
```
Python 3.13.12
```

## Suggested Solutions

### Option 1: Preinstall Artifact Registry authentication backend

Ensure the build environment includes:
```
keyring
keyrings.google-artifactregistry-auth
```
before the pip install step.

### Option 2: Two-phase dependency install

Modify the Python buildpack to run:
```
pip install keyring keyrings.google-artifactregistry-auth
pip install -r requirements.txt
```

Bắt đầu bằng cách xem xét buildpack google.python.pip được liệt kê và lệnh `python3 -m pip install --requirement requirements.txt` trong builder image được chỉ định. Tái hiện lỗi với requirements.txt mẫu và xác minh các quyền của tài khoản dịch vụ Cloud Build. Được xem là hoàn tất khi một lần triển khai Firebase Functions có thể cài đặt gói riêng tư của Artifact Registry mà không cần vendoring, pipeline tùy chỉnh hoặc containerization.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.