Markov Chain of random variables from a primitive markov chain

I'm a little unsure if my approach here is correct, which is making it difficult to find the limit. Any and all help will be appreciated!

Let be a primitive finite state Markov chain with transition matrix and stationary distribution . Define the process by:

Show that is a Markov chain, and compute:

What I've done:

So I need to show that is only determined by , ie:

So, suppose a & b are just some numbers. Then:

Plugging in the definition of :

=

Because are primitive Markov-Chains, can I say that

=

because doesn't depend on ? This is basically where I'm stuck :(