
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]


system

Let \( n \) be a strictly positive integer. Let \(\mathfrak{S}_{n}\) denote the set of bijections of the set \(\{1,2, \ldots, n\}\) onto itself. For \(\sigma \in \mathfrak{S}_{n}\), we say that \((i, j)\) (for \(1 \leq i, j \leq n\)) is an inversion of \(\sigma\) if \(i<j\) and \(\sigma(i)>\sigma(j)\). We also say that \(i\) (for \(1 \leq i \leq n\)) is a record of \(\sigma\) if for all \(j\) with \(1 \leq j \leq i-1\) we have \(\sigma(j)<\sigma(i)\) (thus, 1 is always a record). Finally, let \(\operatorname{inv}(\sigma)\) be the number of inversions of \(\sigma\) and \(\operatorname{rec}(\sigma)\) be its number of records.

Calculate the following sum:

\[ \sum_{\sigma \in \mathfrak{S}_{n}} 2^{\operatorname{inv}(\sigma)+\operatorname{rec}(\sigma)} \]

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