<p>An <strong>ugly number</strong> is a positive integer whose prime factors are limited to <code>2</code>, <code>3</code>, and <code>5</code>.</p>

<p>Given an integer <code>n</code>, return <em>the</em> <code>n<sup>th</sup></code> <em><strong>ugly number</strong></em>.</p>

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

<pre>
<strong>Input:</strong> n = 10
<strong>Output:</strong> 12
<strong>Explanation:</strong> [1, 2, 3, 4, 5, 6, 8, 9, 10, 12] is the sequence of the first 10 ugly numbers.
</pre>

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

<pre>
<strong>Input:</strong> n = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> 1 has no prime factors, therefore all of its prime factors are limited to 2, 3, and 5.
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 1690</code></li>
</ul>

class Solution {
public:
    int nthUglyNumber(int n) {
        unordered_set<long long> seen;  // Use long long to handle large numbers
        priority_queue<long long, vector<long long>, greater<long long>> pq;
        
        seen.insert(1);
        pq.push(1);
        
        int count = 0;
        
        while (count < n) {
            long long cur = pq.top();
            pq.pop();
            count++;
            
            if (count == n)  // Check if we found the nth ugly number
                return cur;
            
            long long x1 = cur * 2;
            long long x2 = cur * 3;
            long long x3 = cur * 5;
            
            if (seen.find(x1) == seen.end()) {
                pq.push(x1);
                seen.insert(x1);
            }
            if (seen.find(x2) == seen.end()) {
                pq.push(x2);
                seen.insert(x2);
            }
            if (seen.find(x3) == seen.end()) {
                pq.push(x3);
                seen.insert(x3);
            }
        }
        
        return 0;  // Should never reach here
    }
};

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