Simple continued fractions and geometric mean

A simple continued fraction expansion of $n$ is

$n=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}}$ ,

where $a_0,a_1,a_2,a_3,\cdots$ are integers and $n$ is a real number.

A geometric mean of real numbers are

$\sqrt[b]{a_0a_1a_2a_3\cdots}$

where $,a_0,a_1,a_2,a_3,\cdots$ are positive reals, $b$ is an positive integer and $a_0a_1a_2a_3\cdots$ has to be positive if $b$ is an even integer.

If a simple continued fraction expansion of $\pi$ , $\sin(1)$ and $\gamma$ and take the geometric mean of the infinite $a_n$ s contain inside the simple continued fraction expansion of the given numbers (That is, $\displaystyle \lim_{b \to \infty} \sqrt[b]{a_0a_1a_2a_3\cdots}$ .) then what is the limit answer (Type the answer up to the fourth decimal digit.).