<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>

from typing import List


class Solution:
    def __init__(self):
        self.memo = {}

    def maxProfit(self, k: int, prices: List[int]) -> int:
        return self.rec(prices, 0, 0, 0, k)

    def rec(self, prices: List[int], num_transactions: int, left: int, right: int, k: int) -> int:
        if num_transactions == k:
            return 0
        memo_key = (num_transactions, (left, right))
        if memo_key in self.memo:
            return self.memo[memo_key]

        while right < len(prices) - 1:
            profit = prices[right] - prices[left]
            if profit <= 0:
                left = right
            elif prices[right + 1] < prices[right]:
                res = max(
                    profit + self.rec(prices, num_transactions + 1, right, right + 1, k),
                    self.rec(prices, num_transactions, left, right + 1, k)
                )
                self.memo[memo_key] = res
                return res
            right += 1
        profit = prices[right] - prices[left]
        res = profit if profit > 0 else 0
        self.memo[memo_key] = res
        return res


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