<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 helper(bought=False,i=0,k=2):
            if i==N-1:
                if bought:
                    return prices[i]
                else:
                    return 0
            if k==0 and bought==False:
                return 0
            STORE = f"{bought},{i},{k}"
            if d.get(STORE, None)!=None:
                return d[STORE]
            ans1 = ans2 = 0
            if bought: # option to sell or not
                ans1 = prices[i] + helper(False,i+1,k)
                ans2 = helper(bought,i+1,k)
            else: # option to buy or not
                ans1 = helper(bought,i+1,k)
                ans2 = -prices[i]+helper(True,i+1,k-1)
            d[STORE] = max(ans1, ans2)
            return d[STORE]
            
        N = len(prices)
        d = {}
        return helper(k=k)
        
65705