
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
On a plane, there are fixed points \(A\) and \(B\). For any \(k\) points \(P_1, P_2, \ldots, P_k\) on this plane, there are two points \(P_i\) and \(P_j\) (\(i, j \in \{1,2,3,\ldots,k\}\) and \(i \neq j\)) that satisfy \[\left|\sin \angle A P_i B - \sin \angle A P_j B\right| \leq \frac{1}{1992}\]. What is the minimum value of \(k\)?

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