Define \( g_n (t) = e^{t + e^{t + e^{t + \cdots}}} \), \(n\) times. And \(\displaystyle f(t) =\lim_{n\to\infty} g_n (t) \). Then which of the following is/are true?

**(a)**: \(f : \mathbb R^- \to \mathbb R^+ \).

**(b)**: \(f(-1) = 1 \).

**(c)**: \(f\) is one-to-one.

**(d)**: \(f \) is increasing on \(\mathbb R\).

**(e)**: \(f\) exists for all \(t \in \mathbb R\).

**(f)**: \(f\) is increasing on \(\mathbb R^- \) excluding \((-1,0) \).

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