Ryan is altering Brice’s coin so that one side appears twice as often as the other. He randomly decides to weight either heads or tails more heavily, with equal probability.

Brice flips this modified coin twice, and it lands on heads both times. We need to find the probability that the coin lands on heads in the next flip. This probability can be expressed as $\frac{p}{q}$, where $p$ and $q$ are positive integers and $\gcd(p, q) = 1$. Determine the value of $p + q$.

Let's think step by step and output the final answer within \boxed{}.
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