Point \(B\) is in the exterior of the regular \(n\)-sided polygon \(A_1A_2\cdots A_n\), and \(A_1A_2B\) is an equilateral triangle. What is the largest value of \(n\) for which \(A_1\), \(A_n\), and \(B\) are consecutive vertices of a regular polygon?

(第十五届AIME1997 第6题)