
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 $3n$ cards are numbered consecutively from 1 through $3n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The next $n$ cards form pile $B$ and the last $n$ form pile $C$. Cards are restacked by taking one card alternately from the tops of piles $C$, $B$, and then $A$, respectively. If, after the restacking process, at least one card from each pile occupies the same position it occupied in the original stack, the stack is named magical. If card number 101 retains its original position, find out how many cards are there in this magical stack.

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