<p>There is only one character <code>&#39;A&#39;</code> on the screen of a notepad. You can perform one of two operations on this notepad for each step:</p>

<ul>
	<li>Copy All: You can copy all the characters present on the screen (a partial copy is not allowed).</li>
	<li>Paste: You can paste the characters which are copied last time.</li>
</ul>

<p>Given an integer <code>n</code>, return <em>the minimum number of operations to get the character</em> <code>&#39;A&#39;</code> <em>exactly</em> <code>n</code> <em>times on the screen</em>.</p>

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

<pre>
<strong>Input:</strong> n = 3
<strong>Output:</strong> 3
<strong>Explanation:</strong> Initially, we have one character &#39;A&#39;.
In step 1, we use Copy All operation.
In step 2, we use Paste operation to get &#39;AA&#39;.
In step 3, we use Paste operation to get &#39;AAA&#39;.
</pre>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 0
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 1000</code></li>
</ul>

class Solution {
public:
    vector<vector<int>> dp;
    int minSteps(int n) {
        for(int i=0; i<n; i++){
            dp.push_back(vector<int>(n, -1));
        }
        return DFS(1, 0, n);
    }

    int DFS(int count, int copy_count, int n) {
        if(count == n) return 0;
        if(count > n) return INT_MAX/2;
        if(dp[count][copy_count] != -1) return dp[count][copy_count];
        
        int return_val = INT_MAX/2;
        // copy
        if(count != copy_count) {
            return_val = 1 + DFS(count, count, n);
        }
        // paste
        if(copy_count != 0) {
            return_val = min(return_val, 1 + DFS(count + copy_count, copy_count, n));
        }

        dp[count][copy_count] = return_val;
        return return_val;
    }
};


// AAA
// 3 3
// copy A
// paste AA
// paste AAA

// AAAA
// 4 4
// copy A
// paste AA
// copy AA / paste AAA
// paste AAAA

// AAAAA
// 5 5
// copy A
// paste AA
// paste AAA
// paste AAAA
// paste AAAAA

// AAAAAAAA
// 8 6
// copy A
// paste AA
// copy AA
// paste AAAA
// copy AAAA
// paste AAAAAAAA

// AAAAAAAAA
// 9 6
// copy A
// paste AA
// paste AAA
// copy AAA
// paste AAAAAA
// paste AAAAAAAAA

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