terminal integration not working with iTerm

### 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 15.6.0 (24G84)

### Expected behaviour

Q doctor fixes issue.

### Actual behaviour

Q doctor reports not running latest terminal integration and directs a terminal restart. Restart does not resolve.

### Steps to reproduce

Run q doctor in iTerm zsh session.

### Environment

```yaml

[q-details]
version = "1.13.2"
hash = "e44ee53d76813a5ef107dcd0d6181b089cb5b2f2"
date = "2025-08-08T20:32:47.811878Z (4d ago)"
variant = "full"

[system-info]
os = "macOS 15.6.0 (24G84)"
chip = "Apple M1 Pro"
total-cores = 10
memory = "32.00 GB"

[environment]
cwd = "/Users/USER"
cli-path = "/Users/USER"
os = "Mac"
shell-path = "/Users/USER/.toolbox/tools/toolbox/1.1.2119.0/toolbox-exec"
shell-version = "5.9"
terminal = "iTerm 2"
install-method = "toolbox"

[env-vars]
PATH = "/Users/USER/.local/bin:/opt/anaconda3/bin:/opt/anaconda3/condabin:/Library/Frameworks/Python.framework/Versions/3.13/bin:/opt/homebrew/bin:/opt/homebrew/sbin:/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:/usr/local/share/dotnet:~/.dotnet/tools:/Applications/iTerm.app/Contents/Resources/utilities:/Applications/Visual Studio Code.app/Contents/Resources/app/bin:/Applications/Visual Studio Code.app/Contents/Resources/app/bin:/Users/USER/.toolbox/bin"
Q_TERM = "1.9.1"
SHELL = "/bin/zsh"
TERM = "xterm-256color"
__CFBundleIdentifier = "com.googlecode.iterm2"
```

Beginne damit, das Problem mit `q doctor` und `q restart` in einer iTerm-zsh-Sitzung unter macOS unter Verwendung der gemeldeten Q-Version und Terminalumgebung zu reproduzieren. Verfolge die Diagnose der Terminalintegration und den Neustartpfad; abgeschlossen ist die Arbeit, wenn die Integration als aktuell erkannt wird und `q doctor` keinen Neustart mehr anfordert.

Vom Indexierungsmodell aus dem Issue-Text verfasst.