[feature] Support running background processes

### Checks

- [x] I have searched [github.com/aws/amazon-q-developer-cli/issues](https://github.com/aws/amazon-q-developer-cli/issues?q=) and there are no duplicates of my issue
- [x] I have run `q doctor` in the affected terminal session
- [x] I have run `q restart` and replicated the issue again

### Operating system

macOS 15.3.1 (24D70)

### Expected behaviour

When asking chat to debug a web application that involves running a local server, expected behavior is that chat runs a webserver, runs required checks, makes a decision that website responds as required, stops local server and continues with its decisions.

### Actual behaviour

The current issue happens when "q chat" makes a decision to run a local server in a current terminal window. This sends the current terminal session into a "running server" mode and stops any further actions from q chat as it loses control over the terminal. Typing "y" or using "enter" do not affect anything. When using "CTRL+C", chat loses the chain of thought

### Steps to reproduce

Ask to create a python flask application and then ask chat to debug the website to see if it works.

### Environment

```yaml
[q-details]
version = "1.7.2"
hash = "b6bf5f1675dd7395009e8bc65c1145820671f211"
date = "2025-03-21T01:36:16.66764Z (19h ago)"
variant = "full"

[system-info]
os = "macOS 15.3.1 (24D70)"
chip = "Apple M1 Pro"
total-cores = 10
memory = "32.00 GB"

[environment]
cwd = "/Users/USER/Documents/aws/samples/q chat/personalize-casino-demo"
cli-path = "/Users/USER/Documents/aws/samples/q chat/personalize-casino-demo"
os = "Mac"
shell-path = "/bin/zsh"
shell-version = "5.9"
terminal = "VSCode"
install-method = "unknown"

[env-vars]
PATH = "/Users/USER/Documents/aws/samples/q chat/personalize-casino-demo/venv/bin:/opt/homebrew/bin:/Library/Frameworks/Python.framework/Versions/3.12/bin:/usr/local/bin:/System/Cryptexes/App/usr/bin:/usr/bin:/bin:/usr/sbin:/sbin:/var/run/com.apple.security.cryptexd/codex.system/bootstrap/usr/local/bin:/var/run/com.apple.security.cryptexd/codex.system/bootstrap/usr/bin:/var/run/com.apple.security.cryptexd/codex.system/bootstrap/usr/appleinternal/bin:/Users/USER/.vscode/extensions/ms-python.python-2025.2.0-darwin-arm64/python_files/deactivate/zsh:/Users/USER/.pyenv/versions/3.8.12/bin:/opt/homebrew/bin:/Library/Frameworks/Python.framework/Versions/3.12/bin:/Users/USER/.local/bin"
QTERM_SESSION_ID = "38b2f8a58ec548a8b6e711aa74579a77"
Q_SET_PARENT_CHECK = "1"
Q_TERM = "1.7.1"
SHELL = "/bin/zsh"
TERM = "xterm-256color"
ZDOTDIR = "/Users/USER"
__CFBundleIdentifier = "com.microsoft.VSCode"
```