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

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

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

Input: prices = [9, 7, 4, 1]
Output: 0

Prices only fall, so no transaction is profitable.

Example 3

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.