LeetCode 212 Word Search II - Why Naive Fails

For each word, run Word Search I (DFS from every cell).

With $k$ words and board of size $m × n$, each word search takes $O(m \cdot n \cdot 4^L)$ where $L$ is word length. Total: $O(k \cdot m \cdot n \cdot 4^L)$.

Many words share common prefixes. Can you search for multiple words simultaneously?