\[\large \frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}} + \frac{a_{3}}{a_{4}}+ \ldots+ \frac{a_{n-1}}{a_{n}} + \frac{a_{n}}{a_{1}}\]

Given \(a_{1} , a_{2} , a_{3} ,\ldots , a_{n} > 0\) for \(n > 2\). If the expression above has a minimum value of \(pn\) for \(p\) is a positive integer, then find the value of \(p + 1\).

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