Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac{1}{4}$, then the probability that the experiment stops with head is  $\frac{m}{n}$ (GCD (m, n) = 1). Then the value of  $n-7m$  is equal to _______

$$\begin{gathered} P\left(H\right)=\frac{1}{4},P\left(T\right)=\frac{3}{4} \\ P=P\left(HH\right)+P\left(THH\right)+P\left(THTHH\right)+P\left(THTHTHH\right)+\mathrm{........}\infty +P\left(HTHH\right)+ \\ P\left(HTHTHH\right)+P\left(HTHTHTHH\right)+\mathrm{......}\infty \\ =\frac{1}{16}+\frac{3}{4}.\frac{1}{16}+\frac{3}{4}.\frac{1}{4}.\frac{3}{4}.\frac{1}{16}+\frac{3}{4}.\frac{1}{4}.\frac{3}{4}.\frac{1}{4}.\frac{3}{4}.\frac{1}{16}+\mathrm{.......}\infty + \\ \frac{1}{4}.\frac{3}{4}.\frac{1}{16}+\frac{1}{4}.\frac{3}{4}.\frac{1}{4}.\frac{3}{4}.\frac{1}{16}+\frac{1}{4}.\frac{3}{4}.\frac{1}{4}.\frac{3}{4}.\frac{1}{4}.\frac{3}{4}.\frac{1}{16}+\mathrm{.........}\infty \\ =\frac{1}{16}+\frac{3}{64}+\frac{9}{1024}+\frac{27}{16384}+\mathrm{.......}\infty +\frac{3}{256}+\frac{3}{4^2}.\frac{3}{256}+\frac{3^2}{4^4}+\frac{3}{256}+\mathrm{.......}\infty \\ =\frac{1}{16}+\frac{3}{64}[1+\frac{3}{16}+\frac{9}{{\left(16\right)}^2}+\mathrm{.......}]+\frac{3}{256}[1+\frac{3}{4^2}+\frac{3^2}{4^4}+\mathrm{.......}] \\ =\frac{1}{16}+\frac{\frac{3}{64}}{1-\frac{3}{16}}+\frac{\frac{3}{256}}{1-\frac{3}{16}} \\ =\frac{1}{16}+\frac{3}{52}+\frac{3}{208}=\frac{1}{16}+\frac{15}{208}=\frac{28}{208}=\frac{14}{104}=\frac{7}{52} \\ m = 7, n = 52 \end{gathered}$$