
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
The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed and kept in order to form pile $A$. The remaining cards form pile $B$. The cards are then restacked by alternately taking cards from the tops of pile $B$ and pile $A$, respectively. In this restacking process, card number $(n+1)$ becomes the bottom card of the new stack, and so on, until piles $A$ and $B$ are exhausted. A stack is named magical if, after this process, at least one card from each pile occupies the same position it occupied in the original stack. Determine the number of cards in a magical stack in which card number 101 retains its original position.

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