Three Rooms, Safety, and Death

You have been caught by a gangster who gives you the chance of escaping by playing a simple game. There are 5 rooms, connected in a cycle as \(S-R_1-R_2-R_3-D\). You are blindfolded. For each room you go, you must choose one door randomly and enter it.

If you reach \(S\) you are released, but if you reach \(D\) you are killed. If you start at \(R_1\), the probability that you will survive the game is \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Find \(p+q\).

Note that you have two choices for the door to enter in each room. You can visualize the rooms as vertices of a regular pentagon, which are labeled in order as \(S-R_1-R_2-R_3-D.\)

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