
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 $1,2, \ldots, 20$ are arranged in a circle in a certain order. For each integer $k$, Matthieu counts how many integers less than $k$ are among the 9 integers that follow $k$ in the clockwise direction; he counts this as $A(k)$. He also counts how many integers less than $k$ are among the 9 integers that follow $k$ in the counterclockwise direction; he counts this as $B(k)$. Matthieu notes that $A(k)=B(k)$ for all $k$. What is the number diametrically opposite to 11 on the circle?

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