<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 {
// public:int dp[5*e4+1][5e4+1][2];
public:
vector<vector<int>>dp;
int solve(vector<int>&a,int fee,int index,bool buy)
{
    if(index>=a.size()) return 0;

    if(dp[index][buy]!=-1) return dp[index][buy];
    int pro = 0;
    if(buy)
    {
        int take = solve(a,fee,index+1,0) - a[index];
        int notake  =solve(a,fee,index+1,1);
        pro= max(take,notake);
    }
    else{
             int sold = solve(a,fee,index+1,1) + a[index]-fee;
        int nosold  =solve(a,fee,index+1,0);
        pro= max(sold,nosold);
    }
    return dp[index][buy]=   pro;
}
    int maxProfit(vector<int>& a, int fee) {
        // memset(dp,-1,sizeof(dp));
       dp =  vector<vector<int>>(a.size()+1,vector<int>(2,-1));
           return solve(a,fee,0,1);
    }
};
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