<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 maxProfit(self, k: int, prices: List[int]) -> int:
        
        N = len(prices)

        @cache
        def dp(i, isBuy, t):
            if i == N:
                return 0
            
            if t == 0:
                return 0
            
            maxProfit = float("-inf")
            
            if isBuy:
                buyNowProfit = -prices[i] + dp(i + 1, False, t - 1)
                buyLaterProfit = dp(i + 1, True, t)
                maxProfit = max(buyNowProfit, buyLaterProfit)
            else:
                sellNowProfit = prices[i] + dp(i + 1, True, t - 1)
                sellLaterProfit = dp(i + 1, False, t)
                maxProfit = max(sellNowProfit, sellLaterProfit)
            
            return maxProfit


        dp = [[[0, 0] for _ in range(1 + 2 * k)] for _ in range(N + 1)]


        for i in range(N - 1, -1, -1):
            for t in range(1, 2 * k + 1):
                for isBuy in [1, 0]:
                    if isBuy:
                        buyNowProfit = -prices[i] + dp[i + 1][t - 1][0]
                        buyLaterProfit = dp[i + 1][t][1]
                        maxProfit = max(buyNowProfit, buyLaterProfit)
                    else:
                        sellNowProfit = prices[i] + dp[i + 1][t - 1][1]
                        sellLaterProfit = dp[i + 1][t][0]
                        maxProfit = max(sellNowProfit, sellLaterProfit)
                    
                    dp[i][t][isBuy] = maxProfit
        
        return dp[0][2 * k][1]
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