Systems of Differential Equations · Superchallenge problem

A point moves in the plane under one of two linear systems, switching every $\tfrac{\pi}{4}$ time units: on $\big[0,\tfrac{\pi}{4}\big)$ it obeys $x'=y,\ y'=-4x$; on $\big[\tfrac{\pi}{4},\tfrac{\pi}{2}\big)$ it obeys $x'=4y,\ y'=-x$; and this pattern repeats with period $\tfrac{\pi}{2}$. Each system on its own has only closed (periodic) orbits. The point starts at $(1,0)$ at $t=0$. Find $x(2\pi)$.