Ideal Lowpass Filters
This looks like a homework problem, so it wouldn't help you too much if I gave you the answer. However, the answer to your problem is actually very educational. You need to model your network using software.
Sequence $w[n]$ will be the impulse response of the lowpass half-band filter. And when you multiply a half-band filter's impulse response by $1,-1,1,-1,\ldots$ you obtain a sequence whose spectrum is equal to a highpass filter frequency response whose cutoff frequency is also $\omega_c = \pi/2$. So the combined frequency response of your parallel two-path network will be the sum of the lowpass half-band and the highpass filters' frequency responses. Devansh, what will that combined frequency response look like? So now the final question is: What is the time-domain sequence (the impulse response) whose spectrum is the sum of the lowpass half-band and the highpass filters' frequency responses?