
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
Anaëlle has \( 2n \) pieces, which are labeled \( 1, 2, 3, \ldots, 2n \), as well as a red and a blue box. She now wants to distribute all \( 2n \) pieces into the two boxes so that the pieces \( k \) and \( 2k \) end up in different boxes for each \( k = 1, 2, \ldots, n \). How many ways does Anaëlle have to do this?

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