```python
def new_cache(limit: int) -> dict:
    if limit <= 0:
        raise ValueError("Limit must be a positive integer")
    return {
        'limit': limit,
        'keys': [],
        'store': {}
    }

def cache_write(cache: dict, key: str, value: int) -> list:
    if not isinstance(key, str):
        raise ValueError("Key must be a non-empty string")
    if key in cache['store']:
        del cache['store'][key]
        cache['keys'].remove(key)
    cache['store'][key] = value
    cache['keys'].append(key)
    if len(cache['keys']) > cache['limit']:
        least_recent = cache['keys'].pop(0)
        del cache['store'][least_recent]
    return [least_recent] if least_recent else []

def cache_read(cache: dict, key: str) -> int:
    if not isinstance(key, str):
        raise ValueError("Key must be a non-empty string")
    if key not in cache['store']:
        raise ValueError("Key not found")
    value = cache['store'][key]
    cache['keys'].remove(key)
    cache['keys'].append(key)
    return value
```