
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
Each of \( n > 1 \) girls has a doll with her name on it. Each pair of girls swaps dolls in a certain sequence. For which values of \( n \) is it possible for each girl to end up (1) with her own doll, (2) with another girl's doll?

[For example, if \( n = 4 \), denote girls by \( A, B, C, D \) and dolls by \( a, b, c, d \). The initial position is \( Aa, Bb, Cc, Dd \). If the swaps are in the order \( AB, CD, AC, BD, AD, BC \), then the successive positions are \( Ab, Ba, Cc, Dd \); \( Ab, Ba, Cd, Dc \); \( Ad, Ba, Cb, Dc \); \( Ad, Bc, Cb, Da \); \( Aa, Bc, Cb, Dd \); and \( Aa, Bb, Cc, Dd \). So for \( n = 4 \), (1) is possible.]

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