Always bringing me back

Consider the simple linear map \(\mathcal{L} : x \rightarrow ax+b\) where \(a=4/7\), and \(b = 5\). Starting at some initial point \(X_0\), we move to our next point \(X_1\) according to \(X_1 \rightarrow \mathcal{L}(X_0)\).

Suppose we iterate the map infinitely many times, ending up at \(X_\infty\), what is the value of \(X_\infty\)?

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