```python
def first_broken_rule(root: dict) -> dict:
    if root is None:
        raise ValueError("Root cannot be None")
    
    def check_node(node, key_bound):
        if node is None:
            return {}
        
        if not isinstance(node, dict):
            raise ValueError("Node must be a dictionary")
        
        if len(node) != 4:
            raise ValueError("Node must contain exactly four entries")
        
        key = node.get('key')
        count = node.get('count')
        left = node.get('left')
        right = node.get('right')
        
        if key is None or not isinstance(key, int):
            raise ValueError("Key must be an integer")
        
        if count is None or not isinstance(count, int) or count <= 0:
            raise ValueError("Count must be a positive integer")
        
        left_depth = check_node(left, key_bound)
        right_depth = check_node(right, key_bound)
        
        order_broken = left and key <= left.get('key', float('-inf'))
        order_broken |= right and key >= right.get('key', float('inf'))
        
        balance_broken = abs(left_depth - right_depth) > 1
        
        count_broken = node['count'] != (1 + left_depth + right_depth)
        
        if order_broken:
            return {'path': 'root' + ('/L' if left else '/R'), 'rule': 'order'}
        elif balance_broken:
            return {'path': 'root' + ('/L' if left else '/R'), 'rule': 'balance'}
        elif count_broken:
            return {'path': 'root' + ('/L' if left else '/R'), 'rule': 'count'}
        
        if order_broken or balance_broken or count_broken:
            raise ValueError(f"Node {key} breaks a rule")
        
        return {'path': '', 'rule': ''}

    result = check_node(root, None)
    return result
```