<p>Given an integer <code>n</code>, return <code>true</code> <em>if it is possible to represent </em><code>n</code><em> as the sum of distinct powers of three.</em> Otherwise, return <code>false</code>.</p>

<p>An integer <code>y</code> is a power of three if there exists an integer <code>x</code> such that <code>y == 3<sup>x</sup></code>.</p>

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

<pre>
<strong>Input:</strong> n = 12
<strong>Output:</strong> true
<strong>Explanation:</strong> 12 = 3<sup>1</sup> + 3<sup>2</sup>
</pre>

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

<pre>
<strong>Input:</strong> n = 91
<strong>Output:</strong> true
<strong>Explanation:</strong> 91 = 3<sup>0</sup> + 3<sup>2</sup> + 3<sup>4</sup>
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> n = 21
<strong>Output:</strong> false
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 10<sup>7</sup></code></li>
</ul>

class Solution:
    def checkPowersOfThree(self, n: int) -> bool:
        cache = {}

        powers = [1]
        curr_power = 1

        while (next_power := 3 ** curr_power) <= n:
            powers.append(next_power)
            curr_power += 1

        if sum(powers) < n:
            return False

        def check(remaining: int, powers_used) -> bool:
            if (remaining, powers_used) in cache:
                return cache[remaining, powers_used]
            
            if remaining == 0:
                return True
            
            if remaining < 0:
                return False
            
            found = False

            for idx, power in enumerate(powers):
                if power > remaining:
                    break
                
                if (powers_used >> idx) & 1:
                    continue

                if check(remaining - power, powers_used + (1 << idx)):
                    found = True
                    break

            cache[remaining, powers_used] = found

            return found

        return check(n, 0) 
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