
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

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[ACTION NAME]

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# Your action step 2

# Your action step 3

...

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system
For positive integers  $n,$  let  $\nu_3 (n)$  denote the largest integer  $k$  such that  $3^k$  divides  $n.$  Find the number of subsets  $S$  (possibly containing 0 or 1 elements) of  $\{1, 2, \ldots, 81\}$  such that for any distinct  $a,b \in S$ ,  $\nu_3 (a-b)$  is even.

*Author: Alex Zhu*

<details><summary>Clarification</summary>We only need  $\nu_3(a-b)$  to be even for  $a>b$ .</details>

Present the answer in LaTex format: \boxed{Your answer}
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