
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 are given a balance and \( n \) weights of weight \( 2^{0}, 2^{1}, \ldots, 2^{n-1} \). In a sequence of \( n \) moves, we place all weights on the balance. In the first move, we choose a weight and put it on the left pan. In each of the following moves, we choose one of the remaining weights and add it either to the left or to the right pan. Compute the number of ways in which we can perform these \( n \) moves in such a way that the right pan is never heavier than the left pan.

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