<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 dp(self, prices, day, num_bought, hold):
        global memo
        key = (day, num_bought, hold)
        if key in memo:
            return memo[key]
        if num_bought <= 0:
            memo[key] = 0
        elif hold == False:
            if day == 0:
                memo[key] = 0
            else:
                memo[key] = max(self.dp(prices, day-1, num_bought, False), self.dp(prices, day-1, num_bought, True) + prices[day])
        else:
            if day == 0:
                memo[key] = -prices[day]
            else:
                memo[key] = max(self.dp(prices, day-1, num_bought, True), self.dp(prices, day-1, num_bought-1, False) - prices[day])
        return memo[key]

    def maxProfit(self, k: int, prices: List[int]) -> int:
        global memo
        memo = {}
        return self.dp(prices, len(prices)-1, k, False)
        
        
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