<p>You are given an integer array <code>prices</code> where <code>prices[i]</code> is the price of a given stock on the <code>i<sup>th</sup></code> day, and an integer <code>k</code>.</p>

<p>Find the maximum profit you can achieve. You may complete at most <code>k</code> transactions: i.e. you may buy at most <code>k</code> times and sell at most <code>k</code> times.</p>

<p><strong>Note:</strong> You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> k = 2, prices = [2,4,1]
<strong>Output:</strong> 2
<strong>Explanation:</strong> Buy on day 1 (price = 2) and sell on day 2 (price = 4), profit = 4-2 = 2.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> k = 2, prices = [3,2,6,5,0,3]
<strong>Output:</strong> 7
<strong>Explanation:</strong> Buy on day 2 (price = 2) and sell on day 3 (price = 6), profit = 6-2 = 4. Then buy on day 5 (price = 0) and sell on day 6 (price = 3), profit = 3-0 = 3.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= k &lt;= 100</code></li>
	<li><code>1 &lt;= prices.length &lt;= 1000</code></li>
	<li><code>0 &lt;= prices[i] &lt;= 1000</code></li>
</ul>

class Solution:
    def __init__(self) -> None:
        self.__mem_buy: dict[tuple[int, int], int] = {}
        self.__mem_sell: dict[tuple[int, int], tuple[int, int]]
        self.__prices: list[int] = []
        
    def __dp_buy(self, idx: int, count: int) -> int:
        if (count == 0):
            return 0
        
        if (idx == len(self.__prices) - 1):
            return 0
        
        key = (idx, count)
        
        if (self.__mem_buy.get(key) is not None):
            return self.__mem_buy[key]
        
        result = max(self.__dp_buy(idx + 1, count), self.__dp_sell(idx, count)[0])
        
        self.__mem_buy[key] = result
        
        return result
        
    def __dp_sell(self, idx: int, count: int) -> tuple[int, int]:
        price = self.__prices[idx]
        
        if (idx == len(self.__prices) - 1):
            return (0, price)
        
        key = (idx, count)
        
        if (self.__mem_sell.get(key) is not None):
            return self.__mem_sell[key]
    
        result = None
        next_price = self.__prices[idx + 1]
        
        if (price < next_price):    
            temp = self.__dp_sell(idx + 1, count)
            result = (temp[0] + next_price - price, temp[1])
        
        else:
            result = (self.__dp_buy(idx, count - 1), price)
            
            temp = self.__dp_sell(idx + 1, count)       
            temp = (temp[0] - price + next_price, temp[1])
            
            if (temp[0] > result[0]):
                result = temp
            
        self.__mem_sell[key] = result
    
        return result
        
        
        
    def maxProfit(self, k: int, prices: List[int]) -> int:
        self.__prices = prices
        self.__mem_buy = {}
        self.__mem_sell = {}
        
        return self.__dp_buy(0, k)
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