
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]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


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Let \( V = \{1, \ldots, 8\} \). How many permutations \( \sigma: V \rightarrow V \) are automorphisms of some tree?

(A tree \( G \) on \( V \) is a connected graph with vertex set \( V \) and exactly \(|V|-1\) edges. An automorphism of \( G \) is a permutation \( \sigma: V \rightarrow V \) such that vertices \( i, j \in V \) are connected by an edge if and only if \( \sigma(i) \) and \( \sigma(j) \) are.)

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