<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:
        
        # T[i] = max()
        # 0 => max(T[i-1]][0][1] + price, T[i-1][0][0])
        # 1 => max(T[i-1][0][1], -price)
        # The trick is to subtract when we buy and add when we sell. (Because that will in the end become positive)

        max_transactions_per_day = 1
        dp = []
        for i in range(len(prices) + 1):
            dp.append([])
            for k in range(max_transactions_per_day):
                dp[i].append([])
                for l in range(2):
                    dp[i][k].append( 0 )
        
        dp[0][0][0] = 0
        dp[0][0][1] = -1e10 # You don't have a stock at the start

        for i, price in enumerate(prices):
            buy = dp[i][0][0] - price
            sell = dp[i][0][1] + price - fee
            dp[i+1][0][0] = max(sell, dp[i][0][0]) # Sell or Hold
            dp[i+1][0][1] = max(dp[i][0][1], buy) # Hold or Buy
        
        # print(dp)
        return dp[len(prices)][0][0]



        
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