Arc length parameter relation with fourth derivative

Question

Let non-planar curve $\gamma:I \rightarrow \mathbb{R^{3}}$ with arc length parameter $s$. Find $a,\,b,\,c$ such that $\gamma^{(4)}(s)=a\gamma^\prime(s)+b\gamma^{\prime\prime}(s)+c\gamma^{(3)}(s)$. I tried with inner product:

1. $\langle\gamma^\prime(s),\gamma^\prime(s)\rangle=1$ because arc length parameter
2. Derivative of (1) $\Rightarrow \langle\gamma^{\prime\prime}(s),\gamma^\prime(s)\rangle=0$
3. Derivative of (2)$$\begin{align}&\Rightarrow \langle\gamma^{(3)}(s),\gamma^\prime(s)\rangle+\langle\gamma^{\prime\prime}(s),\gamma^{\prime\prime}(s)\rangle=0\\&\Rightarrow \langle\gamma^{(3)}(s),\gamma^\prime(s)\rangle+(k(s))^{2}=0\\&\Rightarrow \langle\gamma^{(3)}(s),\gamma^\prime(s)\rangle=-(k(s))^{2}\end{align}$$

I stucked on third derivative, so i thought to use Frenet-Serret, but it didn't help. Any idea?