Optimal Stock Execution with Transaction Fee and Cooldown

A proprietary trading strategy at HRT trades a single equity contract across $N$ continuous trading days. You are given an array of daily stock prices prices and a flat transaction fee fee charged on every sale transaction.

Trading Rules:

1. You may hold at most one share of the stock at any time (you must sell your existing share before purchasing again).
2. After you sell a share on day $i$, you are subject to a mandatory 1-day cooldown period on day $i+1$, during which you cannot purchase shares.
3. The flat fee fee is deducted from net proceeds on the day of the sale.

Your Task

Implement max_profit_with_fee_and_cooldown(prices: list[int], fee: int) -> int that determines the maximum cumulative net profit achievable.

Constraints

• $1 \le \text{len}(prices) \le 5 \times 10^4$
• $0 \le prices[i] \le 10^5$
• $0 \le fee \le 10^5$

Examples

prices = [1, 2, 3, 0, 2]
fee = 1
# Day 0: Buy @ 1
# Day 2: Sell @ 3 (profit = 3 - 1 - fee(1) = 1)
# Day 3: Cooldown (cannot buy)
# Day 4: Buy @ 2, sell same day or hold -> profit 1
# Output: 1

prices = [1, 3, 7, 5, 10, 3]
fee = 2
# Buy @ 1, sell @ 7 -> profit 7 - 1 - 2 = 4
# Day after sale is cooldown.
# Buy @ 5, sell @ 10 -> profit 10 - 5 - 2 = 3
# Total profit = 4 + 3 = 7
# Output: 7