<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:
    int dfs(int cur, int buf, int target, vector<vector<int>> &dp)
    {
        if (cur > target)
            return INT_MAX;
        if (cur == target)
            return 0;
        
        if (dp[cur][buf] != 0)
            return dp[cur][buf];
        
        int result = INT_MAX;
        // Copy
        if (buf != cur) {
            int op = dfs(cur, cur, target, dp);
            if (op != INT_MAX)
                result = min(result, 1 + op);
        }
        // Paste
        if (buf != 0) {
            int op = dfs(cur + buf, buf, target, dp);
            if (op != INT_MAX)
                result = min(result, 1 + op);
        }

        return dp[cur][buf] = result;
    }

    int minSteps(int n) {
        ios_base::sync_with_stdio(0);
        cin.tie(0);
        cout.tie(0);

        vector<vector<int>> dp(n + 1, vector<int>(n, 0));

        return dfs(1, 0, n, dp);
    }
};
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