What's the length of the segment?

$\lambda = \dfrac{1}{2}m(\widehat{PMR})$ by tangent chord theorem $\implies m\angle{PQR} = \lambda$ by inscribed angle theorem

$\implies m\angle{PSQ} = 180 - (\theta + \lambda) \implies m\angle{PSR} = \theta + \lambda = m\angle{SPT} \implies$

$\triangle{PST}$ is an isosceles triangle with $\overline{ST} = \overline{PT} = a + 1 \implies$

$(a + 1)^2 = (2a + 1)y$ by tangent secant theorem $\implies y = \dfrac{(a + 1)^2}{2a + 1} = \overline{RT}$

$\implies \overline{RS} = \overline{ST} - \overline{RT} = a + 1 - \dfrac{(a + 1)^2}{2a + 1} = \dfrac{a^2 + a}{2a + 1} = 1 \implies$

$a^2 + a = 2a + 1 \implies a^2 - a - 1 = 0$ and dropping the negative root $\implies$

$a = \dfrac{1 + \sqrt{5}}{2} \approx \boxed{1.6180339887498948}$ .