
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
At what rate does $\mathrm{E} \min \left(\sigma_{2 n}, 2 n\right)$ go to $\infty$ as $n \rightarrow \infty$? (Here, $\sigma_{2 n}=\min \left\{1 \leqslant k \leqslant 2 n: S_{k}=0\right\}$, and we assume $\sigma_{2 n}=\infty$ (or $\sigma_{2 n}=2 n$) if $S_{k} \neq 0$ for all $1 \leqslant k \leqslant 2 n$.)

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