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Let \(f(x) = \frac{x - 5}{cx + 6}\) for \(x \ne -\frac{6}{c}\), \(c \ne 0\).

1. The line \(x = 3\) is a vertical asymptote of the graph of \(f\). Find the value of \(c\).
2. Write down the equation of the horizontal asymptote of the graph of \(f\).
3. The line \(y=k\), where \(k \in \mathbb{R}\), intersects the graph of \(\vert f(x) \vert\) at exactly one point. Find the possible values of \(k\).