A random graph

Probability Level 5

A graph is constructed iteratively according to the following algorithm. The graph starts as a single vertex. With probability pp, the graph stops here. Otherwise, 3 new vertices are constructed and joined to this vertex. If we have a graph with more than one vertex, for each vertex created at the previous stage, 3 new vertices are construed and joined to it with probability 1p.1 - p. We repeat this process until no new vertices are added. When the expected number of vertices in this graph is 100,100, the value of pp can be expressed as ab\frac{a}{b} where aa and bb are coprime positive integers. What is the value of a+b?a + b?

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