
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 \( n > 0 \) be an integer. We have a two-pan balance and \( n \) weights, with masses \( 2^{0}, 2^{1}, \ldots, 2^{n-1} \). We want to place these \( n \) weights onto the balance one by one in such a way that the right pan is never heavier than the left pan. At each step, we choose one of the weights that has not yet been placed on the balance and place it either in the left pan or the right pan, continuing until all the weights have been placed. Determine the number of ways to do this.

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