JMO 1995 Problem 4

The sequence $\{a_1,\ a_2,\ \cdots\}$ is defined by $a_{2n}=a_n,\ a_{2n+1}=(-1)^n.$ A point $P$ moves on the coordinate plane as follows.

(1) Let $P_0$ be the origin, $P$ moves in a distance of $1$ from $P_0$ toward in the positive direction of $x$ - axis, Denote this point by $P_i.$

(2) After $P$ has moved to $P_i,$ it turns $90^\circ$ to the left and moves in a distance of $1$ when $a_i=1,$ and turns $90^\circ$ to the right and moves in a distance of $1$ when $a_i=-1.$ Denote this point by $P_{i+1},$ w here $i=1,2,\cdots.$ Prove that $P$ can't pass on the same segment more than two times.