```python
from typing import Dict, List

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) or not key:
        raise ValueError("Key must be a non-empty string")
    if key in cache['store']:
        cache['store'][key] = value
        cache['keys'].remove(key)
        cache['keys'].append(key)
    else:
        cache['store'][key] = value
        cache['keys'].append(key)
        if len(cache['keys']) > cache['limit']:
            cache['keys'].pop(0)
    return cache['keys']

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