Telescoping product

$A(x)=x+\cfrac { 1 }{ x+\cfrac { 1 }{ x+\cfrac { 1 }{ \ddots } } }$ With $A(x)$ defined as above, where the continued fraction goes on indefinitely, find the value of the infinite product $\dfrac { 1 }{ A(1) } \times \dfrac { 1+\frac { 1 }{ 1 } }{ A(1) } \times \frac { 1+\frac { 1 }{ 1+\frac { 1 }{ 1 } } }{ A(1) } \times \cdots .$ If your answer can be expressed as $\dfrac { a+\sqrt { b } }{ c\sqrt { b } },$ where $a, b, c$ are positive integers and $b$ is square-free, give your answer as $100a+10b+c$.

Bonus: Can you give a closed formula for $\dfrac { y }{ A(y) } \times \dfrac { y+\frac { 1 }{ y } }{ A(y) } \times \dfrac { y+\frac { 1 }{ y+\frac { 1 }{ y } } }{ A(y) } \times \cdots$ when it converges?