Sunny Days in Mathtown

The weather in Mathtown behaves strangely. On each day during the month of March, it is either sunny or raining. If it was sunny on a particular day, it will be sunny the next day with probability \(\frac{9}{10}\). If it was rainy on a particular day, it will be rainy the next day with probability \(\frac{7}{10}\). Suppose the probability that it is sunny on March \(1^{\mbox{st}}\) is \(p\) and that the probability of it being sunny on any given day during the month is identically \(p\). Then \(p\) can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

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