Question: 2018 Winter Final - 22

Alexa and Shelly want to play the game of Monopoly. They use the following recursive algorithm to decide who goes first:

$\WhoGoesFirst(k):$
$\quad \mathbf{if}\ k \geq 1\ \mathbf{then}$
$\quad \quad \text{Alexa rolls the die, let a be the result}$
$\quad \quad \text{Shelly rolls the die, let s be the result}$
$\quad \quad \mathbf{if}\ a > s\ \mathbf{then}$
$\quad \quad \quad \text{print Alexa goes first}$
$\quad \quad \quad \mathbf{return}\ k$
$\quad \quad \mathbf{endif}$
$\quad \quad \mathbf{if}\ a < s\ \mathbf{then}$
$\quad \quad \quad \text{print Shelly goes first}$
$\quad \quad \quad \mathbf{return}\ k$
$\quad \quad \mathbf{endif}$
$\quad \quad \mathbf{if}\ a = s\ \mathbf{then}$
$\quad \quad \quad \WhoGoesFirst(k + 1)$
$\quad \quad \mathbf{endif}$

The ladies run algorithm $\WhoGoesFirst(1)$, i.e., with $k = 1$. Define the random variable $X$ to be the value of the output of this algorithm.
What is the expected value $\mathbb{E}(X)$ of the random variable $X$?