The subjective probability is defined for any of the agent's 3 possible states. Yet you count the decisions in a different state space under a different probability measure. You are basically using the fact that the subjective probabilities are equal. The number of decisions in each of the branches corresponds to the total subjective measure of the associated events, so it can be used to translate to the "objective" measure.

This is not the standard (state space+probability measure+utility function) model. When you convert your argument to the standard form, you get the 1/2-position on the Sleeping Beauty.