Hilbert number for a family of piecewise nonautonomous equations

For family $x^{\mathrm{\prime }}=(a_0+a_1\cos t+a_2\sin t)\mathrm{\mid }x\mathrm{\mid }+b_0+b_1\cos t+b_2\sin t$, we solve three basic problems related with its dynamics. First, we characterize when it has a center (Poincaré center focus problem). Second, we show that each equation has a finite number of limit cycles (finiteness problem), and finally we give a uniform upper bound for the number of limit cycles (Hilbert’s 16th problem).