An integer is called snakelike if its decimal representation \(a_1a_2a_3\cdots a_k\) satisfies \(a_i< a_{i+1}\) if \(i\) is odd and \(a_i>a_{i+1}\) if \(i\) is even. How many snakelike integers between 1000 and 9999 have four distinct digits?

(第二十二届AIME1 2004 第6题)