
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
Let  $n$  be a positive integer. Ana and Banana are playing the following game:
First, Ana arranges  $2n$  cups in a row on a table, each facing upside-down. She then places a ball under a cup
and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,
where each command consists of swapping two adjacent cups in the row.
Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information
about the position of the hole and the position of the ball at any point, what is the smallest number of commands
she has to give in order to achieve her goal?


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