The Fibonacci numbers are defined recursively by the equation
\[ F_n = F_{n - 1} + F_{n - 2}\]for every integer $ n \ge 2$, with initial values $ F_0 = 0$ and $ F_1 = 1$.  Let $ G_n = F_{3n}$ be every third Fibonacci number.  There are constants $ a$ and $ b$ such that every integer $ n \ge 2$ satisfies
\[ G_n = a G_{n - 1} + b G_{n - 2}.\]Find $(a,b)$.
<|im_start|>system
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<|im_start|>user
The Fibonacci numbers are defined recursively by the equation
\[ F_n = F_{n - 1} + F_{n - 2}\]for every integer $ n \ge 2$, with initial values $ F_0 = 0$ and $ F_1 = 1$.  Let $ G_n = F_{3n}$ be every third Fibonacci number.  There are constants $ a$ and $ b$ such that every integer $ n \ge 2$ satisfies
\[ G_n = a G_{n - 1} + b G_{n - 2}.\]Find $(a,b)$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
(4,1)
The Fibonacci numbers are defined recursively by the equation
\[ F_n = F_{n - 1} + F_{n - 2}\]for every integer $ n \ge 2$, with initial values $ F_0 = 0$ and $ F_1 = 1$.  Let $ G_n = F_{3n}$ be every third Fibonacci number.  There are constants $ a$ and $ b$ such that every integer $ n \ge 2$ satisfies
\[ G_n = a G_{n - 1} + b G_{n - 2}.\]Find $(a,b)$.
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