
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
Fiona has a deck of cards labeled 1 to \( n \), laid out in order from 1 to \( n \) from left to right. Her goal is to arrange them into a single pile through a series of steps in which, if at some stage the cards are in \( m \) piles, she chooses \( 1 \leq k < m \) and arranges the cards into \( k \) piles by:
- Picking up pile \( k+1 \) and putting it on pile 1,
- Picking up pile \( k+2 \) and putting it on pile 2,
- And so on, working from left to right and cycling back through as necessary.

She repeats this process until the cards are all in a single pile, and then stops. 

For example, if \( n=7 \) and she chooses \( k=3 \) at the first step, she will have the following three piles:

\[
\begin{array}{lll}
7 &   &   \\
4 & 5 & 6 \\
1 & 2 & 3 \\
\end{array}
\]

If she then chooses \( k=1 \) at the second step, she finishes with the cards in a single pile ordered 6352741 from top to bottom.

How many different final piles can Fiona end up with?

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