
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 integers \( 1, 2, \ldots, 20 \) have been arranged around a circle in a certain order. For each of these integers \( k \), Matthieu counts how many integers less than \( k \) are among the 9 integers that follow \( k \) when moving clockwise around the circle; he calls this count \( A(k) \). He also counts how many integers less than \( k \) are among the 9 integers that follow \( k \) when moving counterclockwise around the circle; he calls this count \( B(k) \). Matthieu notes that \( A(k) = B(k) \) for every \( k \). What is the number diametrically opposite to 11 on the circle?

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