Let the incircle of \(\Delta ABC\) touches the sides\(BC,CA,AB\) at \({A}_{1},{B}_{1},{C}_{1}\) respectively.The incircle of \(\Delta{A}_{1}{B}_{1}{C}_{1}\) touches its sides of \({B}_{1}{C}_{1},{C}_{1}{A}_{1},{A}_{1}{B}_{1}\) at \({A}_{2},{B}_{2},{C}_{2}\) respectively and so on.

If \(\displaystyle \lim _{ n\rightarrow \infty }{ \angle A_{ n } } = { p }\)

And In \(\Delta {A}_{4}{B}_{4}{C}_{4}\), the value of \(\angle A_4\) is \({q}\).

Then \({p+q}\) can be expressed in its simplest form as \(\dfrac{{m}\pi+{n}A}{{r}}\), where \(m,n\) and \(r\) are integers.

What is \({m+n+r}\)?

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