<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:

        def get_value(x, y, z, profit_matrix):
            if x < 0 or y < 0 or z < 0:
                return -float('inf')
            else:
                return profit_matrix[x][y][z]

        profit_matrix = []
        for day in range(len(prices)):
            profit_matrix.append([[-float('inf')] * 2 for _ in range(2 * k + 1)])

        # profit matrix is day * transaction count * 2 (0/1 whether holding the stock or not)
        profit_matrix[0][0][0] = 0
        profit_matrix[0][1][1] = -prices[0]


        for day in range(1, len(prices)):
            for t in range(2 * k + 1):
                # profit_matrix[day][t][0] = max(profit_matrix[day - 1][t -1][1] + prices[day], profit_matrix[day - 1][t][0])
                # profit_matrix[day][t][1] = max(profit_matrix[day - 1][t -1][0] - prices[day], profit_matrix[day - 1][t][1])
                profit_matrix[day][t][0] = max(get_value(day - 1, t -1, 1, profit_matrix) + prices[day], get_value(day - 1, t, 0, profit_matrix))
                profit_matrix[day][t][1] = max(get_value(day - 1, t -1, 0, profit_matrix) - prices[day], get_value(day - 1, t, 1, profit_matrix))
        

        max_profit = 0
        for day in range(len(prices)):
            for t in range(2 * k + 1):
                for hold in [0, 1]:
                    max_profit = max(max_profit, profit_matrix[day][t][hold])

        return max_profit
                
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