<p>You are given an 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>fee</code> representing a transaction fee.</p>

<p>Find the maximum profit you can achieve. You may complete as many transactions as you like, but you need to pay the transaction fee for each transaction.</p>

<p><strong>Note:</strong></p>

<ul>
	<li>You may not engage in multiple transactions simultaneously (i.e., you must sell the stock before you buy again).</li>
	<li>The transaction fee is only charged once for each stock purchase and sale.</li>
</ul>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> prices = [1,3,2,8,4,9], fee = 2
<strong>Output:</strong> 8
<strong>Explanation:</strong> The maximum profit can be achieved by:
- Buying at prices[0] = 1
- Selling at prices[3] = 8
- Buying at prices[4] = 4
- Selling at prices[5] = 9
The total profit is ((8 - 1) - 2) + ((9 - 4) - 2) = 8.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> prices = [1,3,7,5,10,3], fee = 3
<strong>Output:</strong> 6
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= prices.length &lt;= 5 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= prices[i] &lt; 5 * 10<sup>4</sup></code></li>
	<li><code>0 &lt;= fee &lt; 5 * 10<sup>4</sup></code></li>
</ul>

class Solution:
    def maxProfit(self, prices: List[int], fee: int) -> int:

        m1 = {}
        
        def get_max_profit(index, buyorsell):
            if index == len(prices):
                return 0
            
            if (index,buyorsell) in m1:
                return m1[(index,buyorsell)]

            if buyorsell == "B":
                profitifbuy = -prices[index] + get_max_profit(index+1,"S")
                profitifnotbuy = get_max_profit(index+1,buyorsell)
                m1[(index,buyorsell)] = max(profitifbuy,profitifnotbuy)
                return m1[(index,buyorsell)]
            else:
                profitifsell = (prices[index]-fee) + get_max_profit(index+1,"B")
                profitifnotsell = get_max_profit(index+1,buyorsell)
                m1[(index,buyorsell)] = max(profitifsell,profitifnotsell)
                return m1[(index,buyorsell)]
        
        return get_max_profit(0,"B")
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