Max Profit from Selling Twice

You are given a list of integers `prices` where `prices[i]` is the price of a given stock on day `i`. Write a function `max_profit_twice(prices: list[int]) -> int` that returns the maximum profit you can achieve from **at most two** transactions. A transaction consists of buying one share on one day and selling it on a later day. You may engage in at most two such transactions, but you cannot buy before selling (i.e., you must sell before you can buy again). You may choose to do zero, one, or two transactions. **Function Signature:** ```python def max_profit_twice(prices: list[int]) -> int: ``` Return the maximum possible profit. If no profitable transaction is possible, return `0`. Implement the function so that it handles large input sizes efficiently. The solution should run in O(n) time and O(n) or O(1) extra space.

```python
>>> max_profit_twice([3, 3, 5, 0, 0, 3, 1, 4])
6
>>> max_profit_twice([1, 2, 3, 4, 5])
4
>>> max_profit_twice([7, 6, 4, 3, 1])
0
>>> max_profit_twice([1])
0
```