Convert from sum of products to product of sums (Boolean algebra)

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I had to simplify a boolean expression with a k-map then put it into a NOR-gate implementation circuit. I haven't made the circuit yet, but here is the work I've done:

Original function:

$$F(w, x, y, z) = \bar{w}\bar{x} + \bar{y}\bar{z} + \bar{w}y\bar{z}$$

My simplified function: $$f = \bar{w}\bar{z} + \bar{y}\bar{z} + w\bar{x}$$

Now here's where I got confused - my professor wasn't exactly clear on how to do this. I assume that to put this sum of products function into a NOR-gate implementation, I have to convert it to a product of sums function by finding the complement?

As a result of that, I took the complement of my simplified function above: $$\bar{f} = (\bar{w} + \bar{z})(\bar{y} + \bar{z})(\bar{w} + x)$$

Are my steps and assumptions correct? I've had to basically self-teach myself this stuff so I have no idea if I'm on the right track or not.

Thanks for any help, I appreciate it :)