Let \( X(e^{j\omega}) \) represent the discrete-time Fourier transform of a 4-length sequence \( x[n] \), where \( x[0] = 1 \), \( x[1] = 2 \), \( x[2] = 2 \), and \( x[3] = 4 \). \( X(e^{j\omega}) \) is sampled at \( \omega = \frac{2\pi k}{3} \) to generate a periodic sequence in \( k \) with period 3, where \( k \) represents an integer. Let \( y[n] \) represent another sequence such that its discrete Fourier transform \( Y[k] \) is given as \( Y[k] = X(e^{j\omega}) \) for \( 0 \leq k \leq 2 \). The value of \( y[0] \) is ________ (in integer).
