The figure below shows a board game where the player starts at square $1$ and repeatedly rolls a die to know the number of steps to

advance $($the number of steps is a random integer between $1$ and $6).$ The rules of the game are as follows:

The player wins if he reaches square $6$

If the player reaches square $2$ he automatically advances to square $3$

If the player reaches square $5$ he automatically goes back to square $4$

If the number of steps to advance is invalid, then the player loses. For example, if the player is currently at square $3$ and the die roll is $5$

then the player loses $($as there is no square $8).$

The transition matrix is given as follow: