<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)
        dp = [[[0] * 2 for _ in range(k + 1)] for __ in range(n + 1)]
        for i in range(n-1, -1, -1):
            for transactions_remaining in range(1, k + 1):
                for holding in range(2):
                    if holding == 1:
                        # if holding we have 2 cases we sell or do nothing
                        dp[i][transactions_remaining][holding] = (
                            max(prices[i] + dp[i + 1][transactions_remaining - 1][0],
                                dp[i + 1][transactions_remaining][holding]))
                    else:
                        # if not holding we have 2 cases we buy or do nothing
                        dp[i][transactions_remaining][holding] = (
                            max(-prices[i] + dp[i + 1][transactions_remaining][1],
                                dp[i + 1][transactions_remaining][holding]))
        return dp[0][k][0]
        
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