Roll, roll, roll the dice

A strip of 41 squares is numbered \(0,1,2,\ldots,40\) from left to right and a token is placed on the square marked \(0\). Pinar rolls a pair of standard six-sided dice and moves the token right a number of squares equal to the total of the dice roll. If Pinar rolls doubles, then she rolls the dice a second time and moves the token in the same manner. If Pinar gets doubles again, she rolls the dice a third time and moves the token in the same manner. If Pinar rolls doubles a third time she simply moves the token to the square marked \(36.\) The expected value of the square that the token ends on can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b?\)

Details and assumptions

Rolling doubles occurs when the two dice each have the same number showing.

Regardless of the outcome of the first roll, Pinar "moves the token right a number of square equal to the total of the dice roll".

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