add cmux terminal integration (https://www.cmux.dev/)

### 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 26.3.1 (25D2128)

### Expected behaviour

auto-complete suggestions should appear after running `kiro-cli integrations install input-method`

### Actual behaviour

auto-complete suggestions DON'T appear in cmux
auto-complete suggestions DO appear in Ghostty and other terminal clients

### Steps to reproduce

_No response_

### Environment

```yaml

[q-details]
version = "1.27.2"
hash = "aa2f8e7241504f598fd58ed79f4fee5ab9bf77ae"
date = "2026-03-09T22:57:15.885689Z (2d ago)"
variant = "full"

[system-info]
os = "macOS 26.3.1 (25D2128)"
chip = "Apple M2 Max"
total-cores = 12
memory = "64.00 GB"

[environment]
cwd = "/Users/USER"
cli-path = "/Users/USER"
os = "Mac"
shell-path = "/bin/zsh"
shell-version = "5.9"
terminal = "Ghostty"
install-method = "brew"

[env-vars]
PATH = "/Users/USER/.antigravity/antigravity/bin:/Users/USER/Library/pnpm:/Users/USER/.local/state/fnm_multishells/38837_1773248372300/bin:/opt/homebrew/Cellar/influxdb-cli/2.7.5/bin:/Users/USER/Library/Application Support/fnm:/opt/homebrew/Cellar/docker-credential-helper-ecr/0.7.1/bin:/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:/opt/pmk/env/global/bin:/Applications/cmux.app/Contents/Resources/bin:/Users/USER/.local/bin"
QTERM_SESSION_ID = "d494ad681b784be4b4fed3d123578c0b"
Q_SET_PARENT_CHECK = "1"
Q_TERM = "1.27.2"
SHELL = "/bin/zsh"
TERM = "xterm-ghostty"
__CFBundleIdentifier = "com.cmuxterm.app"
```