Maximum stock trading profit with at most two buy-and-sell transactions

You are given a list `prices`, where `prices[i]` is the price of one share of a stock on day `i`. You may complete at most two transactions, where a transaction means buying one share and later selling it. Return the maximum total profit you can achieve. ### Function Signature ```python def max_profit_two_transactions(prices: List[int]) -> int: ``` ### Rules - You can hold at most one share at a time, so you must sell before buying again. - A sale and the next purchase may happen on the same day. - Fewer than two transactions, or none at all, is allowed; if no trade makes money, return `0`. ### Constraints - `1 <= len(prices) <= 10^5` - `0 <= prices[i] <= 10^5` ### Examples Example 1 ```text Input: prices = [2, 5, 1, 4, 9, 3, 7] Output: 12 ``` Buy at 1 on day 2 and sell at 9 on day 4 for 8, then buy at 3 on day 5 and sell at 7 on day 6 for 4. Example 2 ```text Input: prices = [9, 7, 4, 1] Output: 0 ``` Prices only fall, so no transaction is profitable. Example 3 ```text Input: prices = [1, 2, 3, 4, 5] Output: 4 ``` A single transaction from 1 to 5 already achieves the maximum; a second one adds nothing.