Conway's Conundrum

This sequence is called a " Look & Say " sequence, a type of notational sequence, which obeys the following rule:

$1:$ Start with a seed value $a_0$ .

$2:$ Read the number of groups of consecutive digits (i.e. $121123$ is read "one $1$ , one $2$ , two $1$ s, one $2$ , one $3$ ").

$3:$ Write down the resulting digit (i.e. $1112211223$ ).

$4:$ Repeat.

$312211$ can then be written as "one $3$ , one $1$ , two $2$ s, two $1$ s", which gives $13112221$ .

Notice that when $a_0=22$ , any output $a_i=22$ .

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Note: This sequence was studied by John Conway, who discovered, among many other characteristics, that when the sequence approached $\infty$ and that the seed value $a_0\neq22$ , $\frac{L_{n+1}}{L_n}$ equaled a constant $\lambda \approx 1.303577269...$ , where $L_i$ is the number of digits in the $i$ th element; $\lambda$ is called Conway's constant .