Cognito amplify_backend_manager_APPID deleted

### Before opening, please confirm:

- [X] I have checked to see if my question is addressed in the [FAQ](https://github.com/aws-amplify/amplify-hosting/blob/master/FAQ.md).
- [X] I have [searched for duplicate or closed issues](https://github.com/aws-amplify/amplify-hosting/issues?q=is%3Aissue+).
- [X] I have read the guide for [submitting bug reports](https://github.com/aws-amplify/amplify-hosting/blob/master/CONTRIBUTING.md).
- [X] I have done my best to include a minimal, self-contained set of instructions for consistently reproducing the issue.

### App Id

arn:aws:amplify:eu-west-1:770204761019:apps/d3azouarb564gt

### Region

eu-west-1

### Amplify Hosting feature

Access control

### Describe the bug

I deleted the amplify_backend_manager_APPID userpool in Cognito by mistake.
So I have :
- lost the access on the Studio.
- lost the possibility to push through the CLI which require a connection on the Studio.
- lost the possibility to manage the access to the Studio from aws.console/amplify.

### Expected behavior

The main aws profile still being the same, the possibility to recreate the a new userpool manually or using automatic process through the CLI would be helpful and to link it to the current project.

### Reproduction steps

1. Create an aws account.
2. Follow the AWS amplify tutorial.
3. Delete the automatically created pool on Cognito.
4. Try to access the Studio or to push

### Build Settings

_No response_

### Additional information

_No response_

Start by reviewing the Cognito user pool and Amplify Studio/CLI access flow for the listed app and region. Determine whether a deleted pool can be recreated and linked to the existing project. Done means a supported recovery path restores Studio access, CLI pushes, and access management through the AWS console.

Written by the indexing model from the issue text.