So I have an idea of conditioning on a new set variable say y that is 0 if the first toss is tails and 1 if heads although I am not sure how to structure this answer. I believe the law of total expectation is the most useful formula to solve this although any direction would be greatly appreciated.
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After the first toss, you'll always have a dependency on the last toss. If the last toss is $H$ or $T$, let the expected number of tosses forward be $E_h, E_t$ respectively. We'll write two recursive equations:
For $E_h$, we the next toss is Heads (with prob. $p$) it's finished. If not, we have an expected number of tosses while the last toss is $T$:
$$E_h=p\times 1+(1-p)\times E_t$$
Similarly, for $E_t$, we have:
$$E_t=p\times E_h+(1-p)\times E_t$$
Finally, $$E[N]=1 + p\times E_h+(1-p)\times E_t=1 + E_t$$
