A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by
$$a _ { 1 } = k , \quad a _ { n + 1 } = 4 a _ { n } - 7 ,$$
where \(k\) is a constant.
- Write down an expression for \(a _ { 2 }\) in terms of \(k\).
- Find \(a _ { 3 }\) in terms of \(k\), simplifying your answer.
Given that \(a _ { 3 } = 13\),
- find the value of \(k\).